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PRACTICAL 

BOOK-KEEPING 



AND 



ARITHMETIC. 



DESIGNED 



FOR THE USE OP SCHOOLS, BUSINESS MEN, ETC. 

WITH A 

PRACTICAL AND SIMPLE SYSTEM OF PENMANSHIP, 

AND 

iaxms at %i% grafts, %tmi^% &l 

THE WHOLE RENDERED PLAIN AND EASY BY FULL EXAMPLES. 

BEING 

THE APPENDIX TO THE ANALYTICAL GRAMMAR AND 

DICTIONARY, 



J 
By E. F. DAVIS. 




PHILADELPHIA: 
J. B. LIPPINCOTT & CO. 

1859. 






Entered according to Act of Congress, in the year 1858, by 

E. F. DAVIS, 

in the Clerk's Office of the District Coiirt of the United States for the District of 
South Carolina. 



CONTENTS. 



PAGE 

Addition of Vulgar Fractions 94 

Addition of Decimals 100 

Agreement of Decimals and Vulgar 

I Fractions 105 

( A.Circle or Wheel, Area 122 

( Amount, time, and rat( per cent. 

given, to find the pi-incipal 71 

I Ape. or Account Rendered 18 

A Right- Angled Triangle 123 

' Annuities, Limited and Unlimited 112,113 
I Arithmetical Progression 118 

Barter 63 

Bills of Exchange 12 

Bills Sold 17 

Blackboard, and Construction on Single 

Rule of Three 48-52 

Board or Scantling Measure 123 

Cash-Book 35,86 

Cents Multiplied by Cents 106 

. Compound Interest 73 

Compound Fellowship 81 

Confession 17 

Cotton Sold, and Bill 18 

Cube Root 117 

Day-Book 23-28 

Discount, Commission, &c 75 

Division of Vulgar Fractions 100 

Division of Decimals 102 

Double Position 110 

Double Rule of Three 54 

Drafts and Checks 11, 12 

Due-Bills 10 

Duties, Specific and Ad Valorem 62 

Endorsements 15 

Equation of Payments 65 

Explanation of Pennigrams 2, 3 

Form of Will 20 



PAGE 

Fractions 83 

General Day-Book 38,39 

Gain Per Cent 78 

Geometrical Progression 120 

Index to Ledger 22 

Interest, Simple 65-71' 

Interest, Compound 73,74 

Involution 115 

Inventory on Commencing Business... 23 

Land Measured by the Rod 121 

Ledger 29, 34 

Loss and Gain 78 

Multiplication of Vulgar Fractions.... 97 

** by Cancelling 97 

*' of Decimals 101 

Notes, Joint and Seal 9 

" Negotiable or not 8 

" at Bank 10 

" Judgment 16 

Position, Single 108 

Practice 57 

Present Worth of Money 76 

Principal Amount and Time given, to 

find the Rate per cent 72 

Principal Amount and Rate per cent. 

given, to find the Time 73 

Punctuation, Parts of Speech, and 

Capital Letters 7 

Receipts in General, Legal 13,14 

Reduction of Fractions, Improper, 

Mixed, and Compound 80-89 

Return of Cotton Sales 19 

Rule of Three in Fractions 52 

Sales-Book 37 

Scantling or Joist Measure 124 

(iii) 



IV 



CONTENTS. 



PAGE 

Single Kule of Three.. 46-52 

Square Root 115 

Subtraction of Vulgar Fractions 98 

Table of Fractions 46 

Table of £s. d 57 

Table of Days 107 

Tare and Tret 60 

To find the Least Common Multiple... 89 

Terms used in Business 4 

Titles and Abbreviations 57 



PAQB 

To find a Common Denominator 89 

To Reduce Fractions to a higher de- 
nomination 89 

To Reduce Fractions to a less de- 
nomination 90 

To measure a Crib 125 

To find the Number of Shingles neces- 
sary to cover a roof 126 

Weights and Measures 44 

Their different Uses 45, 46 



ARITHMETICAL CHARACTERS. 



This = denotes equal ; as, 100 cents = $1. 

" — " less, or Subtraction ; as, 8 — 3=5. 

" — " by, or Division, (divided by ;) as, 12 — 6 = 2. 

" -{- " P^us, more, or Addition ; as, 5 -|- 5 = 10. 

" X " Multiplication ; as, 4 x 4 = 16. 

<' : : : : " Proportion ; as, 5 : 10 : : 20 : 40 

" \/ " Square Root ; as, ^ 64 = 8. 

*' ^ " Cube Root ; as, ^ 64 = 4. 



VALUE OF FOREIGN COINS, 



Pound Sterling $4.44.4 

Pound of Ireland 4.10.0 

Pagoda of India 1.94.0 

Tale of China 1.48.0 

Mill-rei of Portugal 1.20.0 

Ruble of Russia 0.66.0 



Rupee of Bengal $0.55.5 

The Guilder of the Netherlands 0.39.0 

Mark Banco of Hamburg 0.33.5 

Livre Tournois of France 0.18.5 

Real Plate of Spain 0.10.0 



! 



APPENDIX. 



PENMANSHIP. 



There is nothing that would seem to demand our attention more 
than penmanship ; although many students not only leave our Pri- 
mary Schools without even making a tolerable progress in penman- 
ship, yet many also leave the College without being proficient enough 
to write a readable hand. In fact, we have often been forced to 
conclude to write an i^?2readable hand seemed to be desired, as it 
would be likely to hide any defects in orthography. Why or how 
this is, we do not pretend to solve ; yet, when, we remember that 
penmanship is the first and legal record among individuals, and also 
among nations, we are at a loss to account for the general yet indif- 
ferent interest manifested by the many. Our teachers must be at 
fault. Are they not, in some degree, responsible for the little ad- 
vancement and interest manifested by the students? The teacher 
should show a lively interest to advance his pupils in penmanship, and 
render regular services in this part of his profession. By industry 
he can render valuable services in this department; and for your 
benefit, as well as that of others, we will now present you with a 
series of examples, as a base for improvement in penmanship, which 
can be executed on the blackboard and slate. The slate is the better 
for training in penmanship ; and no student should allow himself to 
pass beyond the Rule of Three, without making progress therein. 

While in your calculator, learn to wi'ite also ; it is the proper and 
most suitable tipie of all your studies. The learner in arithmetic, in 
executing or solving a question, necessarily has to use certain marks 
in the progress of the question; and now, allow us to inform you 
that those arithmetical marks are the very best base for penmanship. 
Your divisor mark is just the best base known or presented for cut- 
ting the capital letters; and so soon as you cut a good divisor mark, 
you can cut a good letter, as the example under the divisor mark 
will demonstrate. 



"A PENMANSHIP. 

EXPLANATIONS OF PENNIGRAMS. 

No. 1 is executed on the slate by cutting a straight line across 
the slate, and so continuing with as much evenness as you can, 
dropping off at each end until the mark is quite short; then com- 
mence lengthening until you cross the slate, and so on until you can 
mark a straight line, observing also that every line you cut in solving 
a question in arithmetic, is cut straight and even. 

No. 2 is the running of the letter o, which should be practiced 
until you are enabled to make the o round and smooth ; also, until 
you can perform with ease and rapidity, observing to have both the 
top and bottom of the o's on a line. 

No. 3 is a good exercise to train the hand. You may call it "run 
the snake." This exercise should be performed until you can run 
both right and left, retaining an equal and uniform distance in thai 
curves. 

No. 4 represents the divisor mark ^y not changed to a letter. 

No. 5 is the divisor mark, changed into twenty-three different let- « 
ters. All the capitals, except (7, 0, Q, and U, are included ; and Mj 
they may be included, though it does not cut them so well as the 
other letters. 

Now the twenty-three letters, as under No. 5, were changed from 
this base, or divisor mark, as represented under No. 4. 

The teacher should use the blackboard, and the entire school 
should be conveniently seated, with slates,. Let the teacher give 

out. Cut base, t_y or divisor mark; change base into Q^ ; cut 
divisor, <_/ or base; change base into ty& ; cut base ?_/; change 
base into Q/j ; cut base ^y ; change base into ^ ; cut base «_/ ; 

change base into ^, and so on, until all the letters have been so 

formed. Rub out, and proceed again for several times^ — the teacher 
comparing and judging between the pupils as to execution; and, on 
trial, you will improve on each effort. These exercises should be 
practiced by the entire school, as often as once a week. 



Lessons on Hie Slale. 
1 



k^- 



-^—i^-^v i^~i'—i, — iz—^j- 



1^" zr~z-—v 








m ^'/ Ota 



^^% f 




Nole. Never cram/t i/je /la^^d or /iffcjers. M:re/?/ Ac^t/ /Ae/ie/i 



PENMANSHIP. 6 

No. 6. — The first line under No. 6 were o's changed to double 
letters, as explained by the second line. 

No. 1, in first line, was first an o ; then changed to a ; then to a 
and i. 

No. 2, ; then d ; then d and t. 

No. S, to d; then to d and l. 

No. 4, to c^, then to d and 5. 

No. 5, to ^ ; then ^ and j. 

No. 6, to ^ ; then g and / and y. 

No. 7, q and ^. 

No. 8, q and A. 

No. 9, q and A;, and so on. 

It is very obvious, in these examples, that one letter is included 
in another letter, as o, a, and i, o, d, t, etc., are in the same letter; 
so with the divisor mark as a base, the base being nearly correctly used 
in all the letters ; and he who will diligently apply himself in the use 
of those examples, and in extending their changes, will effect in a 
short time more than a life-time would effect in the usual mode. 
Children will readily/ learn to write on this system; and the teacher 
will effect more good in devoting a part of an evening each week to 
his school on this method, than he could during all his time on the 
old plan of giving copies. 

So soon as the school has become organized to this system, the 
teacher can cut letters in the air with a reed or straw, showing the 
physical bearing and feeling in each lesson. This physical bearing 
will grow with the teacher, by noticing and observing his own feeling 
in the cut of letters. The learner or the teacher can change one 

letter to that of another in this way: say. Cut base eV ; change 
base into Q^ ; change Q^ into ^y&; rub out the right side part 

of QS, and the base «_y is left; then cut ^^, and you form fy^; 

and so on, changing the last letter cut into the next letter by rub- 
bing out all bi^^the base, which is readily done on the slate or black- 
board. 

No. 7 includes the continued m, j\ and inverted y's, which are good 
training lessons. 

No. 8 contains different cuts or flourishes, which often give a finish, 
more especially after a signature. They also train the hand well. 



4 TERMS USED IN BUSINESS. 

Observation. — When the pupil has progressed on the preceding 
lessons, he should be supplied with a bound, two-quire Day Book and 
a Ledger, (one may answer — the two are preferable,) and draw off all 
the following forms of Abh^eviatiojis, Bills, Receipts, Drafts, Notes, 
etc., observing to title each, as in the examples following ; also, strike 
lines with a ruler, similar to those presented in the examples. We 
will first present you with abbreviated terms used in business ; and 
in copying, you should retain them in the mind, or at least you must 
refer to them, until you become familiar with them. 



TERMS USED IN BUSINESS. 



Act. or Ape, account. 

Aug., August. 

Amt., amount. 

ApL, April. 

Agt., agent. 

@, at or price. 

B.B., Bill Book. 

Br., brig. 

Bbl., barrel. 

Bot., bought. 

B.P., bill parcels. 

Bts., bolts. 

C.B., Cash Book. 

C.I.B., Commission Invoice Book. 

Co., company. 

Com., commission. 

Cn, credit. 

Dr., debtor. 

Dr. or Doc, Doctor. 

Dis., discount. ^ 

Doz., dozen. 

Dup., duplicate. 

Drft., draft. 

Drge.,, drayage. 

D.S., days. 

End., endorse. 

Efft., effects. 

Ent., entered. 

E.E., errors excepted. 

Feb., February. 

Folio, page. 

Frght., freight. 



LB., Invoice Book. 

Ins., insurance. 

Inst., this month. 

Int., inventory. 

Int., interest. 

Jan., January. 

L.B., Letter Book. 

L.F., Letter Folio. 

Leg., Ledger. 

Mr., Mister. 

Mrs., Mistress. 

Messrs., two or more. 

Man., manufacture. 

Mar., March. 

Mo., month. 

N.B., particular. 

No., number. 

Nov., November. 

Oct., October. 

P., page or pipe; pp., pages or 

pipes, 
Pd., paid. 

Per, by or through. 
Pr., pair. 

Pro, in favor of, ^for. 
Ps., piece. 

P.S., added. (See Dictionary.) 
Pt., pint. 
Reed., received. 
Rect., receipt. 
S.B., Sales Book. 
Sen., older. 



TITLES AND ABBREVIATIONS. 



Sund., more tlian one thing. 

Ult., last month. 

Con, for contra, against. 



Gal., gallon. 
Hhd., hogshead. 
Wht., weight. 



Note. — The end of the little finger, and joint-bone at the wrist, 
should act as a balance in writing, without any pressure on the 
slate or paper ; leaving the motion of the hand free and. easy. 



TITLES AND ABBHEYIATIONS. 



A.A.S., Fellow of the American 

Academy of Sciences. 
A.B., Bachelor of Arts. 
Abp., Archbishop. 
A.D., the year of our Lord. 
Ala., Alabama. 
A.M. has three constructions : 

1. Master of Arts ; 2. Before 

Noon ; 8. In the year of the 

world. 
Apr., April. 
Atty., Attorney. 
Bart., Baronet. 
B.D., Bachelor of Divinity. 
B.V., Blessed Virgin. 
C. or Centum, a hundred. 
Cant., Canticles. 
Capt., Captain. 
Chap., chapter. 
Col., Colonel. 
Cr., 'credit. 
Cwt., hundredweight. 
Chron., Chronicles. 
Cor., Corinthians. 
Conn, or Ct., Connecticut. 
C.S., Keeper of the Seal. 
C.P.S., Keeper of the Privy Seal. 
Clk., Clerk. 
Cons., Constable. 
Cts., Cents. 

D.D., Doctor of Divinity. 
D.V., Deus volenSy God willing. 
Dea., Deacon. 
Dec, December. 



Del., Delaware. 

Dept., Deputy. 

Deut., Deuteronomy. 

Do. and Ditto, the same. 

E., East. 

Ed., edition or editor. 

E.G., example. 

Eng., England, English. 

Eph., Ephesians. 

Esa., Esaias. 

Ep., epistle. 

Esq., esquire. 

Etc., and so forth. 

Ex., Exodus, example. 

Exr., Executor. 

Feby., February. 

Fr., France, French. 

F.R.S., Fellow of the Boyal 
Society (Eng.) 

Gal., Galatians. 

Gen., General. 

Gent., gentleman. 

Geo., George or Georgia. 

Gov., Governor. 

H.S.S., Fellows of the Histori- 
cal Society. 

Heb., Hebrews. 

Hon., Honorable. 

Hund., hundred. 

H.B.M., His Britannic Majesty. 

Ibid., in the same place. 

i.e. {id est,) that is. 

id, the same. 

Ind., Indiana. 



6 



TITLES AND ABBREVIATIONS. 



Is., Isaiah. 

Jan., January. 

Jas., James. 

Jac, Jacob. 

Josh., Joshua. 

Jun., Junior. 

K., King. 

Km., kingdom. 

Kt., Knight. 

K.G.C., Knight of the Grand 

Cross. 
K.G., Knight of the Garter. 
L.C., Lower Canada. 
L. or Ld., Lord or Lady. 
Lev., Leviticus. 
Lieut., Lieutenant. 
Lond., London. 
Long., longitude. 
Ldp., Lordship. 
Lat., latitude. 
Lou., Louisiana. 
LL.I)., Doctor of Laws, 
fib., pound. 

L.S., Place of the Seal. 
M., Marquis. 
Maj., Major. 
Mass., Massachusetts. 
Math., Mathematics. 
M.B., Bachelor of Medicine. 
Matt., Matthew. 
M.D., Doctor of Physic. 
Md., Maryland. 
Me., Maine. 
MS., manuscript. 
MSS., manuscripts. 
N., North. 
N.B., take notice. 
N.C., North Carolma. 
N.H., New Hampshire. 
N.J., New Jersey. 
N.S., New Style. 
N.W.T., Northwestern Territory. 
N.Y., New York. 
Obj., objection. 
Obt., obedient. 



O.S., Old Style. 

Pari., Parliament. 

Pa. or Penna., Pennsylvania. 

Per cent., by the hundred. 

Pet., Peter. 

Phi]., Philip or Philippians. 

Philom., a lover of learning. 

P.M., Postmaster; also, after 

noon. 
P.O., Post Office. 
P.S., Postscript. 
Ps., Psalm. 
Pres., President. 
Prof., Professor. 
Q., question, queen. 
q.d., as if he should say. 
q.l., as much as you please. 
q.s., a sufficient quantity. 
Regr., register. 
Rep., representative. 
Rev., Reverend; Revelation. 
Rt. Hon., Right Honorable. 
R.I., Rhode Island. 
S., South: shilling. 
S.C., South Carolina. 
St., Saint. 
Sect., section. 
Sen., Senator; Senior. 
Sept., September. 
Servt., servant. 
Serj., Serjeant. 

S.T.P., Professor of Theology. 
S.T.D., Doctor of Divinity. 
8S., to-wit; namely. 
Tenn., Tennessee. 
Thos., Thomas. 
U.C., Upper Canada. 
U.S.A., United States of America. 
V. or Vide, see. 
Va., Virginia. 
Viz., to-wit; namely. 
Vt., Vermont. 
Wm., William. 
Wp., Worship. 
Yd., yard. 



PUNCTUATION 



ARKS, ETC. 



PUNCTUATION MARKS. 



, Comma. 


- Hyphen. 


; Semicolon. 


— Dash. 


: Colon. 


H^^ Index. 


. Period. 


* Asterisk. 


? Interrogation. 


t Dagger. 


! Exclamation. 


X Double dagger. 


' Apostrophe. 


§ Section mark. 


" " Quotation., 


II Parallel. 


( ) Parenthesis. 


i[ Paragraph. 


[ ] Brackets. 


A Caret. 


PARTS OF 


SPEECH. 


N. stands for Noun. 


Ad. stands for Adverb. 


Pro. " Pronoun. 


Adj. " Adjective. 


V. " Verb. 


Part. " Participle. 


V.A. " Verb Active. 


Prep. " Preposition. 


V.N. " Verb Neuter. 


Conj. " Conjunction. 


V.T. " A^erb Transitive. 

"V T '^ Vprh Tr»fr«n«;il-ivp 


Int. " Interjection. 



CAPITAL LETTERS USED. 

1. The first word of every sentence should begin with a capital 
letter. 

2. The titles of honor, and proper names. 

3. The appellations of the Deity. 

4. The first word of every line in poetry. 

5. The pronoun /, and the interjection 0, should always be 
capitals. 

6. Cities, towns, villages, seas, rivers, mountains, lakes, ships, or 
any important word, should begin with a capital. 

7. The beginning of a book, chapter, section, sentence, note, or a 
change in the subject, should begin with a capital. 



Note. — The above are presented in order that the learner, in 
copying them ofi", may have them settled in his mind, as it is not 
uncommon for them to be overlooked in the usual course. 



FORM OF NOTES, DRAFTS ETC. 



NEGOTIABLE NOTE. 



dud MJcUau, vauce uceived, ^a/muc^^ ^U^ /(?^7. 



NEGOTIABLE BY ENDORSEMENT. 



/ro flay ^o q{). ^Wtnyo o^ ou/e^, S^ia> <^i^undted 
M)aUau, valooe received, Tanua^'u ^^d^ ^§5"/. 



NOT NEGOTIABLE. 



^ .JYifiG manmd a^^e^ dale, ^ A^amide 'to 

fiay. lo J. 01. 0ioiindliee, €ne S^dau^and ^o/= 
-J^M, vaiue received, ^ed ^yid, /S5y. 



FORM or NOTES, DRAFTS, ETC. 9 

JOINT NOTE. 

rr 'foo tJt(/e€ve 07icnm6 aile^ dafe, i(/e Aio^iii^e 

aicd a7id (oian^'u — ^Daiiatd, vauie ^eceii}ecl, ^^etf^. 



. ^. 



avid. 



laom^e 



JOINT SEAL NOTE . 

/o /lau /o c^ tJ. Qwane^ic^ o^ veoAe'b, ^n/iee 'LJn(}ic= 
6a?id &J)(yii.at6, vatioe 'i^eceived. Jfiine^d ou'i 

na^idd and deaA 'tnid td^m. SA^n^, ^§57 . 

(^aa.y a^'SBee. f'^J.J 

^*j^ Any note becomes a seal note by attaching the seal before assigning. 



NOTE ON DEMAND. 



IT ioo t^n demand, ?_y Aio^mi^e ^o ^ay. ^ 

^ 'fOO 

lamed <^. 



10 FORM OF NOTES, DRAFTS, ETC. 

NOTE AT BANK. 



'^e^^ ^. c^.j do-t a^a/ue deceived, ^^^alcn. ^/^ 



NOTE BEARINa INTEREST FROM DATE. 



^a'^ ^o Jr. Q/b. S97iim> 0^ ^eate^^ .J^ine (S^£un= 
d'ted ^Doiuvu, vauoe secerned, laUd i'jUeiedl ^tcm 
dale, ^6a^cd Sid, /85y. 

*** Any Note or Due Bill will draw interest by inserting <'with interest from date" 
or the time at which it is to draw interest. 



DUE BILL. 



MJu.e c:^. <^. SniUd, cm de^?zand, One Sr4oic= 
(ya/nd M^a^a^d, ^a/ae t'ecemed, ^e/. Syid, /857, 



FORM OF NOTES, DRAFTS, ETC. 11 

DUE BILL PAYABLE IN MERCHANDISE. 

^Due on aco7iand, lo ^aJi^. cn^^iecyn (yjie^u/lcm^ 
^Jntee c/^andied ^Jjo^au 171 Tne'Uy/ianaid'e. 

*^* Any Note or Due Bill may be paid in trade by inserting the articles to be paid. 

DRAFT AT SIGHT. 



Q/wl Maul, ^lazi- ic c/. 6^. cf^'Uted o'i aul&h, r^ive 
^y(:iondted ^IjoMatd^, vatue ieceivedj and cnatae lAe 
^a7?^e lo ^;2j^ accouTU. 

DRAFT AFTER SIGHT. 

S^fiatla^M^ ^ia., e^/ 3d, ^85"/. 
'J^i'il^ day 6 a^&r-' uaAl, ^lau Ic (oet Kyfwiue^ 
c^)-- oidc7'^, Uno '^noio^and ^^ot/atA, 't^auce 'iecei'i^ed, 
d o/talae Ine daynie /<? accotcnl of 

oui^, ^c.j t^wa^mcna 0dc(/aid6j 

CH^aMandiUa' &j)i64uclj Cf. ^ 
y. <S^. ^Jy^cno/6, ^o/tomMa^ S^. ^ 



a7i 



^^ FORM OF NOTES, DRAFTS, ETC. 

CHECK OR D RAFT. 
(o. Q. c/^icc/d^an at ^oaUi., Jfine ^tmJisc/ and 



^a^an. 
}0 



BILLS OF EXCHANaE, FORE IGN. 

&xc4an^e jfaT- ^/OOO. \ 

'1/diU^j. c/a^.6 aj!le7- u^d^ cy/ idid^lul c^ &x= 
c4a/?i^.e, ^6ec(m(/ one/ idlic/ 'i(/?^f^aic/J Aau ^ //^ 

o^^ €n6 3do€c^a77j 0^o€(/iul6 9ie^ina, t^a/ae i.e= 
re^Awrl, and cAal^e ide d^a/me ia acco^mU c^ 

^^ba^a?^ A .^ymata^an. 

J^,,* All Bills, Notes Drafts, Due Bills, Receipts, &c., executed iu a city, town or 

i; s;n!;t7l ;' "'"'f^ "' '^V'^^ ^'"^'" ''^^°"'^' '^^^ ''' ''' ^« ^^ ^^^« «°^"try, except 
at some noted place, they may be drawn without noting the place 

.iJn!' ""I^TTk"^'" *""''• ^" ^^"' ""l Exchange to execute three sets in every particular ' 

nm ar, with the exception as in the example given in' the other two, say,-fir8t and 

third, and first and second, unpaid. ^' uiei d-uu^ 






FORM OF NOTES, DRAFTS, ETC. 13 

PARTIAL RECEIPT. 

4a'c^^, on acccan/^. c/a'?7?.a€4' ^yfkiiytd. 

RECEIPT IN FULL. 



^'W'^iy ^. C7W., Tan'u ^m, ^85"/. 



^^''^ecertvei 



QDot'tatd, tn "/udt o^ a€€ ae^nanad ^o aaie. 



RECEIPT, MONEY PAID BY A THIRD PERSON. 

0hececved o-i Cad. ^. cTwov-v-'u, In'toua^n Jr. 
^. ^'ie^cti^cm, Une ^yw^undted ^Do't'taU, in 4a.M' od 
adt de^nandd aaain6^ lad. <=^. c^r&o-td-u kJi io 
Mi6 dale. Q/b€€ted U^cyued'Cyn, 



RECEIPT FOR NOTE BY ATTORNEY. 



0irecet^^ed, Cane ^5lA, ^856, <(^(mi W{dua7?z 
fjfunne^, ^^ co^eclto7i, One ,JYo-le an Q/h. .j. 
du^oan^d ^(yr- td^tve c7Jcu7?ydted QDattau, dice ^ycc/n. 
^U. ^856. ^a'Vo, 0d2c^au/6 ^ ^aucj^i/e, 

^cytneud. 



14 FORM OF NOTES, DRAFTS, ETC. 

RECEIPT AFTER SUIT. 



^r^. ^/unne^ \ ^^^^ >^ ^-^^4 /^^/; ^T^O 

RECEIPT ON THE SAME BY THE SHERIFF. 

"W^a/cam Wztnne^ \ 0lecemecl, q4?u^. SOid, ^§57, 
^^- ) o/ ^e/l., t^mee cy^tondtecl and 



^' ^^^^^^^- ) Si^/Uu^^^ ^jyuolia^d, m w^ Oi 



ce, 0^unclAai'j tynle^he^l, and ^o-d-id in /^^ 
^(/l€/^ln ca^e. Q/h. Jfinaoj 



ORDER FOR MONEY. 

iman ^. ^.; Jicne yii, /Ssy. 
'" a 



^?ya7^ lo &I)a4>id 
S^emfiMan uicol^y^ QI)cl/ai6 on demand, and 
ona^ae in.6 d-ame ^o i^e accaan^ o4 



OiCU, 



oUn Kyf(o ^(^la^'u. 



FORM OF NOTES, DRAFTS, ETC. 15 

ORDER AND RECEIPT. 



cy^aAlinad Une (::y^K/?uAed and U^^c/en^u &£) 044^6, 
aha I /lid '}?Z'U otaeT 6Aa€€ t^e 'uoa^r^ lecei^U €0"?^ €ne 
Aa'?7ie. 

Kj^amme^. 



la/r^t utamTni 



ENDORSEMENT, (RESPONSIBLE.) 



/:o &/). ty. A ^. tJ^f/ii/uj and aua^aniee in^e ^^a'u= 



ENDORSEMENT, (NOT RESPONSIBLE.) 



io cyw. ^. f^{)a6on, ^a■i^AoiU iecaat^-e ^c 'r}^y.^6^^ (y^ 
4du, ^di6 SFe^, 6^d, 4857, 



nneT^. 



16 FORM OF NOTES, DRAFTS, ETC. 

JUDGMENT NOTE. 



^'U ^ox (yr auler^, ^rie Mem o4 tJif/c U^^ato^a/nd 
&D(>t^aA6, M(;U'U da'ud ^'lo-o')^ aaie ; and <_y n&'tet^'U 
no-^nincUe, can^^i^ioie, and a^iAai^i^ tne d^aid €)/i 
^ox, or a/?vu a^^o^ne'U=al=€a/i/j' o-€ i^id uiaie, ^rmf 
&oiie and la/Kj-^t' ailo'ineT/^^ tu.evoca/v-€e, 4o^ one a/}^d 
m m'u name ^o aAhea/r^ m an/i/. cou'U o€ ^iyecatd a€ 
^Aid Creole J ai anu lime adle^ Ine av^ave A^amU'= 
^o■tu no-le t^eca^me^ dae, and lo w-aiv-e aiZ A'baced'd a/nd 
^eWice Irie/oeod, and lo can^e^d ladameotl €>■€ Ine 
AoMe^ Tie^bea^ dryr i^e MM7Z Inal 7?tau w due and 
ot(/ina neteon, ^(/Un. iniete^l and ca^^, and ^(^alv^= 
ma atZ e'b'iou, ^c. 

J/n ^{/il'ne^6 ^tf■fle^6ee^, tV neieionto ^ei o^zy^ 'Uand 
and A-eat^, in lAe ^£Di6''^ucl ojf ^Aalian/uaw and 
c/ta^e ad CfowtA ^aiouna, ^nid /(§ /^ day. o^ la^', 
^857. 

/ 0^. 0^oi^nd^ue, ^^.Sr.J 

C/ianed m Ine ^le^ence o4 
KJmti 



FORM OF NOTES, DRAFTS, ETC. 17 

CONFESSION. 



ca^e, and c(y?i€e66 juaa^ienl ^o^t &ne ^'nou^ana 
a?ta KJia>lu=(>n6 ^JjattaUj w^i^n. inleted^ €ta97z uefi= 
^enme'i Sy^n-j ^S55, a/?td condenl IncU juaa^ien^ ve 

0. Q, ^hicama/n. 



BILL SOLD. 



t^^9'-. lad. (ybna&hdGn 

/ ^a/e ^e-m/i ^a^^iii^, 900 yc/^. @ SO f 0^^0.00 

A ^^aci.^ ^c/fcc, 'f50, 'f60, 430, 490=560 /A @ 40 f 56.00 



.Gji- 



^yiec d ^aume^j 



fier 3~.W^. mmd6. 



18 



ACCOUNT RENDERED. 



a4bl. Cfl'* 



,% jr. 



aiMdj 



^. 



4^56. 



S^d. 



0e 



/7 



t/o 



S flal^ ^/loea, ^.50, S.25 

4 fia. SS^o. cJ&07n6dhi'On, 34 
44 yc/d^. ^ inaAamd , @ 25 f 
S4 yda. ^tini^, @45f 

4 docc. kJ^ luted ^ W7n 



05.00 
3.00 
3.75 



44 



3.M 
3.50 
3.45 
4.50 



4 <i-ei^ & ^anneii o/< 



Q^alance due, 



75 
.50 
.00 



45.00 



44 



26 



45 



75 



56 



56 



00 



56 



COTTON SOLD, AND BILL. 



t^/m^. Ced'd'e zjmd'Ofi 






'^ce 



6 ^aied %GUon, fi^cz. : 

3M, 33^, 372, 3^6, 355, 342 = 2434 Ma. @ 

^ /// S^cc^cm, 240 /L. @ 42f 
'/ aacd Coffee, 450 Ma. ® 46 f 
5 f^yc. S/^Aoea, © §4.60 
S aacia S^ait, ® 04.75 



va'V-6 



24.00 
^.00 
5.25 



^^^^, 



66.05 
.04 



ST. ^. 0ama. 



*^* The teacher should as often as once a week interrogate the pupils on terms used 
in business, abbreviations, notes, drafts, receipts, &c., alternately giving the character 
of the question, and require the student to repeat or solve the like question. 



19 



RETURN OF COTTON SALES. 



Sa/e6 o^ KJnt'U^ ^Da/ed ^ol^on (y?^ ace /. a^ Q4ya= 

E.P- A60, JJ&\ A56, J^52, i5j^, J50, U6 , 



AM, MA, AiS, ii/, AAO, U^ , A68, 

A5^, A56, A54, 450, AA§, AA6, A AS, ) = 43,A56 ® g^^f §4295.U 

AAA, AAA, AA7, AM, AA9, AAO, A50, 

A4^, AA6, AAS 

^aan haici /letant a7id azayage, pS^.OO 

^a<iA ^laid /cT tehainna, y .50 

^lotaae / loeeid, @- 6 f fit lu. 4S.60 

^o»i^ju4^ion on §'f295JA, @ SV^f 32.3^ 9^ M 

€Pcl ^a/. "^aJv i7i fu//, §^203.66 



The learner thus far in these examples should in the back part of his or her book, copy 
them off, observing to rule and give the forms as laid down in the examples ; and by this 
attention you have procured for yourself a knowledge of practical information that 
otherwise you would not have procured. They emphatically present a business know- 
ledge that every man should attain ; provided you have retained in your mind the 
practical bearing of each part according to its nature and use. You no longer are de- 
pendent on a second person to execute your writings of daily occurrence ; you are now 
enabled to do your own business, and assist that heretofore second person, if you 
should meet with one, who has not been fortunate enough to have the advantages 
these forms present. These examples for the learner in the school-room are also em- 
phatically offered, as prescribed by law and the usage of trade, as a guide and reference 
to any and every person connected or associated in business transactions. It will pay 
the merchant, the farmer, the mechanic, and more especially the youth, who, by the 
advantages it offers, will fill our country with practical and business young men. 

It is necessary for persons of any occupation whatever, or who are engaged in bu- 
siness to any extent, to have some method of keeping an accurate record of their 
transactions ; and that method which our most intelligent merchants have tried and 
approved mny fairly be allowed to be the best, and highly necessary to any pursuit of 
life, — as the farmer, mechanic, or any occupation whatever requires the same know- 
ledge in his business as the merchant. What is needed by all occupations is a prac- 
tical work, not a theoretical one, — one which will give the forms of doing business as it 
is actually done in our daily transactions, which has induced the author to present 
this practical work foi- the express purpose in each of its departments, — the practical 
and beneficial results attainable or included in our highest schools, — of elevating the 
common school practically equal to the advantages of the high school. 

It is truly gratifying and commendable to see some of the schools of high reputa- 



20 FORM or NOTES, DRAFTS, ETC. 

FORM OF WILL. 



WMia od ^olcnd 7)iind cond ^jne^natu^ do 'i7ia'R.e and 
AiiMid'fi '^nu on'u ta^l ^w-idZ and ic/Uamenl, in 
Tna/nne^^ and €a^m ^ot/a^a-ma: ^'i^d^l, Jf ai^^e and 
"^eaaealA lonla m'u ^aetat^ed ^(^■t€e, ^. ^£D.^ ^ne /o^= 
ww-ma, ^0 ^(^ii: r a^, onajj^ w.J tyle^/n, J^ acve 
and tf-eaaea^A ^o 77tu etded^ ^on, 0. ^^., me €o-t-€czci= 

tna^ io icAii: Jh^(mt, ^y ame a?id ^eaaealA le 

onu d^on «y. cy^. me ^(yi^a^c^lna^ ^o ^(/U U'iem, 

Q^€ie^ i^e dealA o4 ^nu ^e€(wed ^(/i^e ^. &D., ty 
aive a7td devtde oA^t i^^rie 'te^^ o€ '77Z'u e^lale, ic Se 
divided, m^ ^a^e o^ o■l'£et^a-i^e, Ao ad io ^ende^ ^771^ 
v-ealoeal'n7ne^^l6 eaaa€ a77za7za 071^ a{iot^e=7ta'77ted 
cn.it'die7^. ^y do na77ii7iate and aAAo&nl ^777^7^ t(/i<Ce 
7^ &D . and dan^ &. ^. a/7zd Q. (y^&. ^0 v-e in^e 
eccecaiaid o-i iAid onp tadl ^(/l€Z and ie6la'77tenl. 

J'n ^'^inionu ivneteod^ kJ navs dui^cidea mu name a?7cl 
a4-/txea ni7/- deai^^ mM aay c/ en I//6 ^lea^^ o4 out' 



.tooid 0726 ^Acu<ianco eia/U uu7tdted and- 

tion, that a lively interest is being manifested in favor of, or rather substituting, 
practical education for that of theoretical. In fact, this interest is manifested now in 
many of the best female schools ; throwing off much of the very affected feminine 
education and substituting in place thereof that which is solid and practical, thus pre- 
paring her for usefulness in the position she occupies in real life, rather than that which 
a visionary or deluded fancy would have her to be. 



FORM OF BOOK-KEEPINfl. 21 



DAY-BOOK. 

Accounts of persons are Dr. for each article they buy and for all 
amounts they become responsible for, and are to be Cr. by each 
amount paid in cash or in any other manner. 



LEDGER 

Should include each and every account full to itself on the proper 
folio and indexed correctly corresponding with the alphabet according 
to the first letter of surname. 



CASH-BOOK. 

Cash is Dr. to all money received pertaining to the concern, and 
Cr. by all money paid out, pertaining to the same. 



SALES-BOOK. 

Sales are Dr. to Cash for all goods sold, and Cr. by amount for 
all sold, giving profit and loss. 



These four hooks include the best and most practical system of Book-Keeping by 
Single Entry, and they are preferable to Double Enti'y for all ordinary purposes ; 
more simple and easily understood, — although all persons engaged in mercantile pur- 
suits should use other books as auxiliaries, such as Order-Book, Bill-Book, Invoice- 
Book, Lktter-Book, Check-Book, Receipt-Book, &c., which respectively are only 
used to copy Orders, Bills, Invoices, Letters, Checks, Receipts, &c., "which is often tribu- 
tary to the settlement of any mistakes or litigations that occur : in fact the merchant 
who fails to duplicate his orders, bills, letters, receipts, &c., is liable to confusion in 
business and settlements, if not to losses. 

^:f,* "^Ve now proceed to present the Day-Book, Ledger, Cash-Book, and Sales-Book, 
in the order as above. 



22 



FORM OF BOOK-KEEPING. 



The learner should first copy the examples in the Day-Book, and then post them 
into the Ledger, and then you must use names of your own selection, say some ten or 
twenty names, written in the Day-Book or on a sheet of paper, and use those names 
as your customers, bearing in mind now and then to enter some Cr. to them as paid. 
Also you should vary the articles purchased ; also the Cr. given as much as you can ; 
and you will find yourself more than paid by making the articles purchased fractional in 
quantity, also fractional in price in the calculation of the different articles, as each 
calculation made in this way renders doubly the benefits to a question laid down in 
arithmetic ; it renders to you the very thing education designs, that is, to make you a 
practical man, competent to calculate, write, dictate, &c. for yourself; for, be assured, 
education in any branch is of no avail further than you are prepared to reduce to prac- 
tice ; and to attain to be highly practical approaches the ambition of the most aspiring. 



INDEX TO LEDGER. 



t^^ 



ni^uion, ^ahi. C7. 



^tow, }7tuia/}n 



^ 



S 



$r 



<7& 

"S^iU, /ad. ^^. 



^s-/ 



J^ 



^r.^.^. / 



^ 



■}ck!Scef ^Ua%dt 



7 



kJ zamniei, ^ym. 






0^. 0S. s. 



FORM OF BOOK-KEEPING. 
DAY-BOOK. 



23 



SSudmcAd C^c/iooij i/ ^laz^an'^tt/ia Q^Jt^-^.^ ^an. ^Sln^ ^o5y . 







,y^7ivc7iloi'u 0/ 'fny &//eciii on C07n7ncncina 


ifuMnea^. 










S^/dc, cfQ)6U^ J^ ou^e. 












^^u G^^ecla ai6 aa ^oUozoa : 












^ad^n 071 naTid, 


$3^0.00 










rr. fr. kJ noTna^t ow&a Tne 07z xJroie, 


^90.00 










^6euAa7ic/i<}6 aa ^17-. ^Tivcice^, 


2j6^.9i 










t-/ . T. ^cJl6^e/to7^ ozo&d 97ic oti % 

^. S^. Ky^(}e.ad<yu/a cto64 07z Jroie a7id % 


75.50 






• 




^30.A0 


$3537 


U 






ty owe: 












S^. T. k/ ncmaa on ^yToie fof ^oo^i^e, 


§^5.00 










^^e7iz'u c/wUcceT-, % foo^ ^017^, 
0. '2P. x./^^oiaan, 07z JPole, 
€Pezty ^ozi^nan on %, 
^L'^.^o/efo.'^ro^i, 


4i0.00 
40.50 
30.75 
25.00 










'W'. ^K ^o/e, /or daUn^ ^ood^, 


4i.^0 


343 


05 






p^9^ 


79 









Obseryations. — Any transaction may be entered on the Day-Book, specifying the 
nature of such transaction in "words suitable and of easy comprehension, which can be 
better stated by the nature of the transaction than by examples. 

You should enter in Ledger, Stock or Merchandise Dr. to all goods bought as per 
Invoices, and Cr. by the sales per day, week, or month ; and so doing you know your 
stock on hand. Also, open Loss & Gain account, Loss Dr. to cost of goods, and Gain 
Cr. by sales, and you have the gain. 

■5^^* The Day-Book, Cash-Book, and Sales-Book should each be headed with the 
name of your place of doing business. Name the city, town, village, or place of busi- 
ness, as Columbia, Charleston, Spartanburg Business School, &c. 



24 



FORM OP BOOK-KEEPI]^G. 



DAY-BOOK. 



iycjud 


tne^ 


iii. C7cHaoC^ ^ na^€a7^'0a'^a ^zJidt.j ^ 


ran. 'Jd 


m, /o 


^/. 






(^a/U. C/ 1771 eon c^d le/if to7i , 


Q)^. 










^0 9 yc/^. ^Lii7tc, @ 75 f 


06.75 










„ //^. 4S yci6. ^f^/tUe ^inen, @ 40 f 


A.^0 






1 




„ 4 €Ponyee 9t^c/^f., @ 50 f 
„ i'/.^ S/'/u>6^, @ §4.50 


4.60 
S.OO 
3.00 










„ 3 fz^. ^cuoTz ^'^o^e, @ syy^f 


4.4SV2 


§49 


S7y 










Q)r. ^. 5^ Cf(?i/yoze, 


0^. 


y 






STo e/^yds. ^11 ^'coacUd, © $3y^ 


§6.56% 










„ SVzyrL. ^t^add/7iiGie, @ //% 


4.37y2 






1 




// '^6 yda. ^ i,7iaAa?na f ® SO f 


3.S0 










„ /^K yd^. 9f^ic7to, @ //^ 


43.4 sy 










„ /iK ^/l S^cM, @ //^ 


46.03% 










„ 5 M. ^. ^i?. 0oai>^t7iy^, @ SOf 


4.50 


U 


79y^ 










^azcAy ^i:7c^ce, Od-a., 


00^. 








STc SOOfl. ^i7icLo7' ^/a^a, /^ X /i* @ 6V' 


§46.00 






1 




n / ^ay^a'ua '^l^o/fee, 460 Md. ®45f 
„ / /// 6f/tCZ77t M, 32 yJd. @ 90 f 
„ Soy yda. ^atfi,eiir>y, @ 6S]4 


SA.OO 

s^.io 

37.^4% 




'•■ 




'fA 


n Vi c/ua STca, 30 //^. @ 90f 


S7.00 


433 


64y^ 








fan. 


/a,. ^. 9tf;fC 


0^. 








STo / do^. ^'/aie^, @ SOf 


§S.JO 










,t / Q)a7y-^ook,, A aaiiea. 


4.00 










,, . / J:S^ef/acr, S or-. 


.50 






1 






A.S5 
4.00 










„ H (/o. „ £^cU6T, @ §3.50 

■ 


4.75 


43 


90 









*^* Each individual should be titled according to his rank ; also, Mrs. and Miss to 
those to whom the title belongs. 



FORM OF BOOK-KEEPING. 



25 



DAY-BOOK. 



tyjuji 


?^(?^ 


<:t t7cnoo€^ C^ nalfant'Wia ^2yid€.j ^ 


'^an. 'J /I 


m-, 'J a 


o/. 


1 






0^. 
$45.00 

4A.A0 
5.25 
5.20 

42.00 


§54 

9 
40 

30 

1 

74 


^5 




• 




2 


0r. ^. S. ^offozc^, 
kJ 2 aooi. Afuo/<} ^auaanii7i2, @ 60f 
„ 2 „ „ €Pazeyouc, @ 70 f 
„ 10 M. Sfi^09n ^a/i^, © 6 f 
n / ^•^. f. ^ 6c/, 
„ K „ fiL n „ @ §3.00 


4.20 
4.A0 
.60 
^.50 
4.50 


20 








2 


STo J^SV^ M. S/u^ar-, @ 40 f 


0r. 

04.25 

2.00 

2.25 

.75 

4.25 


50 








2 


0^. S/". ^ooa^u/f, 
^ 20 yc/^. 0^zinU, @45f 
n //^. ^f. ^07ne4fi,vn, 34 yM. @ 44 f 
„ 4 /i4. ^u,n7tu ^aaqi7ia, 60 ycla. @ 20 f 
n 4 coii0^6fte, 54 /^4. % 42<P 
„ A fir. ^wyan 6f/icea, @ §4.50 


0^^. 

$3.00 
3.i4 

42.00 
6.42 
6.00 


53 








2 


^0 93y^ /u^d. ^oz7t, @ 5614 f 
n 34\i aaM %.^om<i^ef} , @ 68^ 


0^. 
§52.73i-^ 
24.4^1-^ 


24% 









26 



FORM OF BOOK-KEEPING. 

DAY-BOOK. 



ud^tedd- <L7cnootj, iryiai^anva/ia SJidtj Jan. ^ Am^ /oS^. 



15 






/aa. =^ 
STc ^SYa.'ucL. ^azfieicna, @ 56Mf 



0^. ^. ^ cf(?ayo'. 

^y e'fV^ 4u^l ^ieai, i 



Joa/ii. C7i7neon Gu^e'u/to'^^. 



> a<^<fi??7eze, 



^0 3% yM. ^a 
n 46yz yc/4. S^iM, @ //j^, 



^^. i'J'i'M^^. ^acon, @ ^mf 






0^. 

A.OO 

3.00 

.15 

S.50 



00 



^.65% 
Sy.90% 



7.03% 



0. 



'16 

S^.65% 



U% 



0.. 
^S0.5A% 

/ojpll 



J3y3 



'it 



AO 



30 



05 



^5% 



3yy2 



1^ 



FORM OF BOOK-KEEPING. 



DAY-BOOK. 



aMnc4<i ^Wiccij ^ ^lai^anvuia ^zJt^^.j Jan. ^/mj ^85 / . 



4^ 



Ji* ^a^i^n on % 



3^0 30 S/'ca^idiny 40/1. /cny, S^yJi-^^ 300 fi., 

@ 50 f $4.50 

n 90 €Piaiii ii fi. /cna, ^ ^u 4 — 840 fi., 

6.7S 



SbV. 



m C7l % 



40 



ly 60 /^/. ^ctoz, @ 60 f 



s&v. 



n S Muga tJ ovauc, 



0r. 

$3.00 

j50 



Tan. ' 49 



0r. ^. ^. a^i/yoze. 
STc 4 U S/'aya'T, 2iO Ma., @ 43f 
n 4 M. ,^6o/a,i^&!}, A5 aa/t}., @ 70^ 

^y U A^/. ^/leat, @ $4.40, 



0^. 

S4.S0 
34.50 



^a/tt. ^ ^i 



te^i/lon. 



%J 4 hr. k/ %ace ^ncLi'tK}, 
n ^ (/c^. ^co/^aia, @ iOf 



0r. 
$4.S5 



00 



36 



00 



50 



70 



05 



28 



FORM OF BOOK-KEEPING. 

DAY-BOOK. 



^udm^dd S^c/iog/, S^fiaUan'^ma Q^mI, Jan. ^Q^d, /§57- 






0^. 

§6 00 

i.50 



.00 



/i^i/ua7}t ^^a9n'm6/ 


(©^'-. 


^y SiO /L. ^eed^ollon, @ Wzf 


09.50 


„ /i ^u^/l. €^ea^, @JOf 


9.^0 


„ ^ dcz. S'oga, @ ^Vs, 


4.00 



SO 



50 



30 



FORM OP NOTES, DRAFTS, ETC. 

LEDGER. 



29 



QJ. 



^^. ^ ^. 



4^57. 

/an. 

JPoi>. 


/J 


■ 


$6S.70, 


i $^07 
43 






04^0 


77^/2 









^^ 




^a^^. Simeon 






4S57. 

/an. 


43 


Q)ay.^ooi f^^e S, 049.S7'A; 0^a^e A, 
^a^e 5, 06.05, 


§m-9^h> 

\ 

i 
1 


1 0^7 mi 

6] 05 




053\ 30\^, 




\ 



QJ. 




^mdzy 






4^57. 
/an. 


43 


Q). ^.fia^e S, 433.6 4y2; 0^. 3, 07A.S4%, 
^ 5, 0^.SS, 


0SO7 

6- 


i3% 




0S46:^ 05% 



0. 



/ad. <tiy. 



4^57. 

/an. 



4 A Q). ^./la^e S, 043.90 J €P. A, 0AO.S5%, 



05. 



45y, 



^^ 




^"leae^cc 






4^57. 
/an. 


4A 


0. ^.fui^e 3, 054.^5; 0. A, 03O.7A\l, 
S/'o ^a/. ^a^i tn ^ett^menf, 


1 
1 

0^S 

A 


59\\ 




0^7 


u\\ 




\ 





■5^^* The teacher of course mil solve those fractional quantities and valuations to the 



80 



FORM OF NOTES, DRAFTS, ETC. 



LEDGER. 



tyLi^t/aote. 



4^5y. 

/an. 




$^20 


77'A 


$^20 


rr'A 







lew^n. 



O. 



&^. 



4^57. 



^(^a. 



i5 



^y ^a^fi tit fu/tl, 



i, $S6.U% 



0S6 
27 


^6h 


$53 


30\\ 



'c^cJee, iO^a. 



'^. 



fan. 



Q). ^./la^e 5, $2^ JO, 
i^y 'itoi^ ^i^o dayd a/iei^ date, 



02 A <fO 
^(j}\ 95% 



6\ 05% 



(O^ 



V^ 



< 4^57 




fan. 


a 


[ODec. 

1 

1 


25 



0. ^. fia^e 5, §20.00, 

t^'u cneoK, on K/Te^iwe^^y■ SSani 



00 
3^ <75% 



■4' ^5% 



<l7 71/1^67^. 



^7 



-CT^. 



i^57. 

/an. 



/5 



0. ^.fia^ A, $5/.Uil; ^. 5, 036.00, 



^-^M 



'■n 



pupil, although the learner had better commence in whole quantities and at valuations 
of familiar and easy count. 



FOEM OF NOTES, DRAFTS, ETC. 

LEDGER. 



31 



^^. 



^^. ^.^: S. 



4^57. 

/an. 



^au^^ooi /mas 3, 09-^0; <2P. 6, §40.50, 



^^. 



Hiiam 



/an. 



Q). ^.fio^e 3, §40.50, 



^. 



: ^ 



/^ 



^./la^e 3, 030.53, 



&r 



cam 



/i 



'taae i, §45.05 ; €P. 5, §3.50, 



70 



10 



50 



30 



§30 



53 



55 



^ Those examples should be copied and separately solved, as those fractional quanti- 
ties and valuations are more minute than are practiced by accountants, book-keepers, 
&c. ; yet they are just such as all accountants should readily understand, and are pre- 
cisely the very thing for the learner in arithmetic ; as he who or the arithmetic that 
does not to the very extent of fractional remainders retain or express their amount 



,32 





LEDGER. 






"m/for^. 


^.. 


4^57. 
JPcv. 


'79 

9 


Q)ay=^eoi /la^c 6, 0<75.OO, 
^y ^ea/'c?nent in % 


§^5 00 

i, 70 


0^9 70 




ZJl(2?7t97ZCt. 


^^. 


'4^57. 
/an. 

\ 

\ 


/j? 


^. ^.fiaye 6, §20.30, 


§so 


30 


$so 


30 






, noomcifp 


&^. 


^^57. 

/an. 


/5 


^y'^a,^in/<J^, 


$30 


53 


^lom. 


%9'-. 


< 4^57. 

\ 


/5 


^y'^aJ>.if^fuJ^, 


//^ 


55 



is incorrect and deficient. For nothing short of the last possible fraction in the teach- 
jing or the solution of a question, is sound or correct teaching; and in the proper 
management of those small fractional parts, and their true conception, is the strong 
arm of mathematics. They present and proffer to the learner the highest and most 
comprehensive view in the science. 



FORM OP BOOK-KEEPING. 

LEDGER— DE. AND CR. ON SAME PAGE. 



QJr. 3J^. ^. ^, 



ouao^e. 



?^. 



/<^57. 



J?o. 



^3 



^0 ^c/^e. 



/o // C/con. 



n (o adrv, 



€P. 


\ 




\4^57. 




s 


§U 


79y^ 


Ja^. 


45 


5 


6S 


70 

49''4 


n 


4^ 


§407 




1 43 


^^Vsl 








1 


1 



C7U 



07S 



syVx 

AO 



77'A 



^^. "^a/a. ^ 



^duw^Tt. 



?^. 



4^57. 



^y C7f.(/?t. 



It n 









4^57. 




2 


049 


S7y^ 


/an. 


45 


^ 


27 


9H-. 


(^cL 


S5 


5 


6 


05 








p3 


30Y% 













It ^a<i-n, 



4jm 



3011 



QJt^. ^atc/^y 



'wctycJeCj (o^c 



^^. 



4^57. 


\ 


/an. 


43 


It 


45 


tt 


47 



t/o 



C/u/yv. 









4^57. 




s 


0433 


64y2 


/an. 


47 


A 


7A 


S4% 


Q)ec. 


7 


5 


1 


S2 






/S46 


05% 







0jif ^a^n, I 5 
„ JPcie, I 



494- 



40 
95% 



05% 



^^. /a.. ^ 



M/l 



W7. 



SToS^u 



II II 



S5y^ 



45% 



4^57. 

90 \/an. 



0e 



II ^ncci 



34 



00 

45% 

45'A 



34 



FORM OF BOOK-KEEPING. 

LEDGER— DR. AND CR. ON SAME PAGE. 



Q^r. Wuc/eiic^ 



i/wUze'T'. 



?^. 



4^57. 






^■i 


1 

1 


fisr. 






0". 


1 


/an. 

It 


/i 


ST'o ^un. 


3 i ^54 ^5 


Jan. 


45 


^y S^u,n. 


Ji. 


$54\U\\ 


45 


n n 


Ui 


30\7£^ 


II 


47\ II ^ctn, 


5 


36 00 


II 


47 


II >> 


1 i 


il^ifli 




1 

1 

1 










«61 


m 


^^7 


im 






1 — 




i 


1 






0^. 0r. a 



anoic/. 



^^. 



4^57. 



4^57. 

Tan. 



^9 



fi :3rc ^f^c/^. j 3 ,1 §9fO i 

/•^ n ,1 \6 \\ 40\50 \JPo'u.\ 9 



70 



il I 

A i 






04 5' 00 

M70 



70 



^^. ^tffuim 



% 



la.^nmet 



^9^. 



Tan. 



/il Jo ^fU<u^. I 3 



50 
^0 i 



4^57. 
Tan. 



30 \ 



49 



^J-u ^u,n. 



30 



$S030 



^^. m. -^ 



-(/lu^. 



4^57. 

Tan. 



44\ STc jfU^^. 3 !j $30 



4^57. 
53 \Q)ec. 

ii 



53 



^y ^a^T, \3\ 030 



53 



03O\53 



Qj.^. 'm/&. 



no2i>. 



Tot 



f^r^. 




See note, page 39. 



FORM OF BOOK-KEEPING. 

CASH-BOOK. 



35 



^zJ'T. ^c^adinedd <L7(ynoo/^ J-anaaZ'u Sc/j /oS^ . 







ty '^adn on hand, 


§U5 


60 






„ n fzo'm ^. ^r. ^a^^n^ on JPote, 


^5 


72 






„ n /zo9n ^. ^. 6/iuiM on % 


Ji-3 


60 






n fio^n ^. ^ ^o/e, frr ^oocU 


7 


^4% 




3 


It ,1 ^^0'm S^. 0. ^u/itte^-, ^oo- ^ooia, 


6 


50 




§S^9 


23)4 


^an. 


<^ ^adn on nand, 


^m 


A^H 






II II /o"/^ C/uqa% a7id ^cuec, 


40 


00 






II n ^z>09n t^^oo%e 3C (^^aaa on % 


// 


50 






II II Aom cdba-mhian ^octaan, 


/ 


50 






,1 n Ao?n /. C7. 0hoqe^■!i on % 


54 


60 




Jf 


II ,1 ^1/mi ^^iMed ^zemlon, 


^0 


50 




^335 


5^'li 




t^ ^adn on nand, 


§255 


03K 






II 1, ^or ^ood^ do/d ^i/aa ^endooz, 


§0 


34y, 






,1 II for ^ooda 4ofd ^. 0. ^eona^d, 


AO 


60 






n ,1 i^O'}n =^. ^ . ^./mieadotc-J 07i % 


34 


20 


ti 




,1 n ^M-m ^aa. ^^oMy on % 


S5 


00 






II II Ao?n YytncAneu '^to'i-a on J\^oie 


i9 


AO 






II 1, for ^ood. ^/d ^r. .0^. ^,/l 


A2 


00 




5 


II II /or i^ coda aotd ^cJa^dl'u■ t^bc^ee, 


23 


56% 




05A7 


40% 




^0 ^a.dn on nand, 


0S73 


63'% 






„ II fzo7n f. 9^. ^c^^an 07z % 


50 


00 






II II /to}7t Ky . / . i:^tei{/io7z, 


27 


60 


It 




„ fro7n ^t/e7^ 06Azd, 


M 


50 






n II Aom fad. ^6a^d07^ 07i Jrofe., 


§50 


00 






II .It AoTn Tad. ^amc/i^t 07i % 


400 


00 




6 


II ,1 Ao77t S^. f. 3^Ao7nad 07V % 


^5 


50 




§^27 


23% 


n 


^0 '^ad^A on 4and, 


§629 


4i 



36 



FORM OF BOOK-KEEPING. 

CASH-BOOK. 



ludt'nedd ^c/ico/j J-anua7/u Sd^ ^o5y . (^. 







^y ^a^^/td. ftety^l £^. ^. 0^oad, 


§6^ 


90 






„ ,1 fui. ^c'T zeftaizata ^Icze, 


47 


50 






,, n fid. /or ^ct7i, 


^3 


60 






„ „ fid. ^or ^^acon and ^lUlei^, 


// 


75 




3 


^3u ^a^n (yn na7id, 


//i 


4^K 




0S^9 


23V^ 


/an. 


^y ^a^A/u/. ^ou'li, 


p 


00 






n n fid. /cT toad c/ Jy cod, 


/ 


50 






„ „ /id. €Pa^ci ^oy en % 


65 


^0 






,, ,, fid. ^of dodder, 


40 


25 




A 


^u ^ad-4 on /land, 


S55 


03V, 




0335 


5^% 


ft 


Sou ^a4n fid. (y . =25. .=:L>ind4e'u, /of ^ott<yn, 


//^/ 


40 






n fd./. ^ ""/Kodicc//, 0,1 JPote, 


40 


00 






,, tt fid. ^0^ Qtoou,4^e=0^en(;, 


50 


00 






n ft fid. /or- &Q<J^ and ^juU&r, 


S 


37y2 




5 


^U ^a<^/t, on nandj 


sy3 


63Y^ 




0547 


40% 




^y ^a^/fd ^^. ^. ^/u^ma,, /or ^ozn. 


070 


^5 






ft ft fid. ^ban^iaiq C^dwatd^, 


60 


AO 






ft ff fid. Jr. Gloa^iV'!/, for c/eed SoUon, 


3<f 


iO 






ft ff fui. famea S^mo/d, /or ^otn, 


35 


50 






c^^y ^ati^n on tiajid, 


6S9 


AiK 




0^27 


63H 






1 






*^* The Cash-Book is used to record the daily receipts and expenditures of money. 
J We have presented you with four days' transactions, and each day's transactions during 
* the year is continued in the same manner. 



FORM OF BOOX-KEEPIJTG. 

SALES-EOOK. 



37 



va^me^^ i/'cnoolj Jan. 2c/^ ^85^. QzJf-. ^ 







§5.00 ,1 


4.50 


„ 0.50 


„ 7-00 


II 7.50 


1 

1 /^^ 


50 










7.50 „ 


S.50 


II 0.75 


II 40.00 


II 40.50 


! 




34 


25 


fan. 


3 


4.50 „ 


3.00 


II 4.00 


II 4.75 




40 


35 






ft 


II 


S.S5 „ 


3.50 


II 6.S5 


,1 S.50 








44 


50 


tt 


A. 


A.OO ,1 


7.00 


II 44.00 


II S4.00 




43 


00 






ft 


II 


5.50 II 


9.00 


II 46.00 


,1 S9.OO 








59 


50 


ft 


5 


0.S5 „ 


0.30 


II 0.60 


n 0.45 


II .0.70 


S 


30 






n 


II 


0.40 n 


0.50 


„ 0.90 


II 0.65 


1, 0.90 






3 


3d 


It 


6 


45.00 „ 


0.40 


II ^.00 


II 4.50 


II 0.05 


S7 


95 






n 


II 


SS.OO n 


0.70 


II 900 


II 7-50 


„ 0.40 






39 


30 


ft 


7 


4.00 II 


S.OO 


II 7.50 


II 4.35 




44 


75 






It 


II 


4.50 n 


3.S5 


1, ^.50 


n S.50 








45 


75 


ti 


i 


0.50 II 


0.60 


„ 0.40 


II 4.50 


II 4.00 


7 


00 






It 


II 


0.70 II 


0.90 


II 0.60 


II S.OO 


,1 6.00 






40 


30 


II 


40 


3.00 II 


7.00 


1, S.50 


II 4.50 


„ S.OO 


49 


00 






II 


II 


A.50 1, 


40.00 


„ 4.50 


,1 6.50 


n 3.00 






S^\50 


n 


// 


3.50 ,1 


6.50 


1, 3.40 






43 


40 






II 


II 


4.75 „ 


9.50 


II 6.00 










20 


35 


It 


/i- 


4.00 II 


4.50 


,1 S.50 


II 3.50 




^ 


50 






n 


'/ 


4.50 II 


S.50 


II 3.50 


II 4.50 








4S 


00 ' 




0464 


65 


0334 


60 










cf cun>, 








464 


65 








069 


95 













This Sales-Book, if correctly kept, will enable the merchant, at the end of each day, 
month, or year, pretty correctly to exhibit his gains ; also it is a very correct Stock- 
keeper. You find in the ten days' sales as above, the cost of the goods which are ex- 
hibited in the Dr. column to be $164.65, and those in the Cr. column exhibit the sale 
of the same to be $234.60, giving a gain of $69.95 : so with each day's trade you are 
enabled to know your gain. Take the 4th, for instance: the cost is $43.00, the sell- 
ing $59.50, giving a differencQ of $16.50 gain. Now, by ledgering the amount of 
goods bought as Dr. and Cr. by each day's sale, you have the means of a daily Stock 
exhibition; for instance, your Stock on 2d January is $3500. Subtract each day's' 
sale, that is the cost price, or any number of days, and you have the Stock : say the 
ten days' sales, which are $164.65, from the Stock $3500, and the Stock on hand is 
known to be $3835.65. 



88 FORM OF BOOK-KEEPING. 

THE BLOTTER, OR GENERAL SALES-BOOK OR DAY-BOOK. 



^Judtncii^ ^ cfiooij <t/ ^iat^aniti7a i::2)i^l.^ /an. Sd^ /q3 / 







^oUJP. ^. Q)avi, 0.. %: 


■ 










3 fu). '^untd, 30, 34V2, 33yz = 95 yJ^. ® p f 
S /i4. ^w. 9i^ome^/iu7i, 3A, 35 = 69 yM. @ t 

/ c/<yz. J:^a(/i64 ^Go<te, 

//^. 0uj/iez4, 30 yd^. @ 20f 

/ /// S^uyaT-, SAO, ® ^Sf 
/ aaci ^of/ee, ^40, @ ///' 
/ /// ^JL/a4<ie4, AS ^a/a. © 6O f 

4 aacia ^ail, @ 0S, 

c/o/a .^3. 75. Q/Jzewicn on ii?n6 kJtoU .' 
/ /t/id. S^uya<r, ^90, Q) 9f 

5 Mcia Coffee, ^A5, ^65^3^/0, @ ^Of 


$^.55 

^f 5.5S 

3.00 \ 

. 3.00 

6.00 

6.00 


pS 

77 

444 


07 




^si.io 

45.40 

S5.S0 

^.00 


40 




^^0.40 
34.00 


40 












4 yo/d ^WaU^daiTz, 


$45.00 

7.50 

6.00 

90.00 


44^ 


50 




$20.00 








4 frie 9'^ae, 
i fir. ^Pant^, 


4.00 
6.00 










'f f/r. c/cui&i,4, 


3.00 


33 


00 



By subtracting each day's sale, (that is, the cost price,) you will know your Stock, 
and so by the week, month, or any numVier of days — say the two days' sales as in the 
i Sales-Book is $164.65 from the Stock $3500 and the Stock is known to be $3335.35. 



FORM OF BOOK-KEEPING. ti 

THE BLOTTER, OR GENERAL SALES-BOOK OR DAY-BOOK. 



^dudtnedd ^ cnoo-i^ 'LTyiat^anmc^.a ^zJmI.^ /an. Sd^ /o3/. 







S'o/d ^,n. ^. Sr/^ma, on % ; 










/ ca^e ^00^^, /i'/.^, @ p. 00, p6.00 
/ ca<i6 ^^c^a'}^ ^/toes , 40 ft^., @ $4.00, 40.00 










40/17-. ^ac^ieJ ^aae^4, @ 04.40, 44.00 










SO /M-. ^ac/ieJ ^fyfi6%^, @ 75 (^ 45.00 


0405 


00 










4 fir. fine c/od 0^ania, 040.00 
4 fzr. C^id ^/ovea, 4.50 
















4 /?/>-. fine ^ooid, 6.00 
4 ^o/ci='^6aded ^ane, 5.00 


22 


50 










C/oul J. S . t^/Mpadcn : 








5 Ud^. S/^u^ur, 910; 9^4, ^70, 4040, 9^S, 

= 4i4S, @ ^f §3^5.04 
4 L^^ ^^afi/ee, 454, 46S, 464, 465 = 642, 

@ 9f 57.7^ 
200 ^a^4 ,9^/101, 25 /I ccid, 5000 M.,®7f 350.00 

6 ^oxea €^aidin4, @ 02.00, 42.00 




i04 


^2 



*^'^' This book is sometimes called the Blotter, the General Day-Book, or Sales-Book. 
It is a book in which several or all in the firm or house may make entries. When a 
sale is on '^, as in the first example, it must be transcribed •)oth to the Day-Book 
and Sales-Book. The second example is sold for cash, and must be posted or copied to 
the Sales-Book and Cash-Book, or, if settled at the time of purchase by note, copy to 
Sales-Book, (that is, the amount, &c.) 

This Blotter-Book may with propriety be used — in fact, it should be used — not only as 
in the examples given, but as a general entry-book for all transactions done in the 
firm, stating the nature of the transaction done, even if of such a nature as need no 
further notice or copying to other books, acting as a note for reference, &c. The 
book-keeper can readily discern all entries of this nature ; also of those that should 
go to other books. 

Note for page 83. — We have presented two forms of Ledger : the first is prefera- 
ble where the customers trade frequently during the year, as the latter would require, 
in a retail trade, one or more entire pages, with many customers dealing as often as 
once a week during the year. It is very proper to enter the Cr. of all settlements on 
the Day-Book, although it is more convenient and also more common to enter the set- 
tlement in full only on the Ledger, as you refer to it in making a settlement. However, 
all partial settlements or payments should be entered on the Day-Book at the time of 
such payment. The partial payments are entered on the Day-Book and Ledger in 
those examples, and the settlement entire on the Ledger only. 



PEACTICAL ARITHMETIC. 



We now present to the learner or the scientific arithmetician a 
system of practical arithmetic; and in so doing, we proffer no advan- 
tages to the skillful in practical calculations ; we propose not to in- 
struct the literary connoisseur in practical arithmetic ; we present 
no attractive graces of style to charm, no daring flight to asto7iish; 
no new researches to gratify ; but we do presume and proffer to 
teachers, parents, business men, and learners, a system of arithmetic, 
in conjunction with a system of Book-keeping, conjointly embracing 
an arrangement of much practical utility. The design of this entire 
work is presented to the learner to learn for himself, and its arrange- 
ment necessarily demands this ; yet rendered, by examples, so sim- 
ple and plain, that none but the inattentive and undiscerning can 
fail to comprehend. We propose, in the entire plan, to make you 
practically equal with the most scientific, not by a general cram- 
ming in, but by a general drawing out; not depending on another, 
but with manliness to rely upon your own powers ; not having to 
depend on this work or that work, but rather to learn and instruct 
you to depend on yourself. We, with all candor to the different 
arithmetics, concede their usefulness ; not, however, without some 
scruples as to their practical simplicity. We further concede the 
correctness of many Ready Calculators and Arithmetical Diction- 
aries; but we do not concede that they present any feature toiuard 
education, hut rather to the reverse — torpify and cinpple practical 
education. 

Candor, learner, requires us to say to you, that you might just as 
well have a second person to eat your dinners to appease your appe- 
tite, as to have a Calculator or Arithmetical Dictionary to do your 
calculating ; and just so long as you allow another person to eat 
your dinners, your appetite will be unappeased; and undoubtedly, so 
long as you rely on Calculators, you will be ignorant and unshillful 
in practical arithmetic, as none are educated who are void of arith- 
metical skill. 

The operative powers of Arithmetic are embraced in four charac- 
ters, viz.: Addition, Midtip)lication, Subtraction, and Division; 
and they preseyit but little difficulty to the learner, as he can readily 
learn to add, multiply, subtract, and divide, as correctly as the most 
skillful; and yet, he is unable to solve any intricate question. Now, 
the learner should inquire why this is so. We will endeavor to point 
you to the mystery. It is not in the operative performance, but in 

(41) 



42 



PRACTICAL ARITHMETIC. 



the operative arrangement and application. The difficulty is in the 
bearing of the question; and here let us remind you of the import- 
ance of the grammatical and analytical advantages of this work, for 
a true grammatical and analytical conception is your only resource 
for the solution of questions. Keep those four operative powers 
{Addition^ Multiplication, Subtraction, and Division,) properly 
arranged, and you will not fail to solve any question. Their proper 
arrangement contains the power and force of all arithmetical science ; 
and not, as you probably suppose, the four operative characters. 
The learner, after he has become expert in Addition, Multiplication, 
Subtraction, and Division, would solve readily any question that this 
work might propound, or that of anybody else, were he as profi- 
cient as they in the bearing of a question; hence the importance to 
the learner to learn to conceive, or to cultivate his conceptive power 
in a constructive sense, and not allow himself to be perplexed because 
he is unable to perceive the bearing of the question in the four ope- 
rative powers referred to above. No, sir ; you must learn the gram- 
matic and analytical force, and the necessity of arithmetical con- 
struction and bearing ; and this word-work greatly assists you in 
this department. It proffers to you the true construction of any 
and each technical term used in Arithmetic, with their true arith- 
metical construction, together with the true grammatical construc- 
tions. An arithmetical problem may be compared to a team of 
horses ; and if each horse is properly equipped, harnessed to his 
proper place, and all drawing together, there is hardly a prospect of 
a stop. Just so with those four powers; work them in \\i.q\v proper 
harness and places, and a failure is impossible. 

MULTIPLICATION TABLE. 



1 

"2 
3 
4 
5 

~6 
7 

~8 
9 

lo 

11 

12 


1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
ll 

12 


2 
4 
~6 
8 
10 
12 
14 
16 
18 
20 
22 
24 


3 

6 
9 


4 

8 


5 
10 


6 
12 

18 
24 


7 
14 
21 


8 
16 


9 
18 


10 
20 
30 


11 
22 
33 
44 


12 

24 
36 

48 


12 


15 

20 


24 


27 


12 
15 


16 


28 


32 


36 


40 


20 


25 
30 


30 
36. 


35 

42 


40 


45 


50 
60 


^^ 


60 


18 


24 


48 


54 


m 


,72 


21 


28 


35 


42 


49 


56 
64 


63 


70 


77 


84 
96 


24 


32 


40 


48 


56 


72 


80 


88 


27 
30 
33 
36 


36 


45 


54 


63 


72 


81 


90 


99 


108 
120 


40 
44 

48" 


50 

^^ 

"60" 


60 


70 


80 


90 


100 


110 


m 

72 


77 


88 


99 


110 


121 


132 


84 


96 


108 


120 


132 


144 



PRACTICAL A R I T H :.I E T I C. 



43 



WEIGHTS AND MEASURES. 



What are the denominations of 
Federal Money? 
10 mills make 1 cent. ct. 
10 cents " 1 dime. d. 

10 dimes '' 1 dollar. $ 



What are the denominations of 
Wine Measure ? 



10 dollars 



eagle. 



E. 



What are the denominations of 
Sterling Money? 

4 farthings 1 penny, d. 
12 pence 1 shilling, s. 

20 shillings 1 pound. £. 

5 shillings 1 crown. 

21 shillings 1 guinea. 

What are the denominations of 
Troy Weight? 
24 grains 1 pennyweight, pwt. 
20 pwts. 1 ounce. oz. 

12 ounces 1 pound. ft). 

What are the denominations of 



Apothecaries' 
20 grains (gr.) 

3 scruples 

8 drachms 
12 ounces 



Weight? 

1 scruple. 9 

1 drachm. 5 

1 ounce. S 

1 pound. ft) 



What are the denominations of 
Avou'dupois Weight? 
16 drams (dr.) 1 ounce. oz. 



16 ounces 
25 pounds 

4 quarters 
20 hundrdw't. 


1 pound. 
1 quarter. 
1 hundredw't 
Iton. 


ft). 

qr. 

. cwt. 

T. 


What are the denominations of 
Dry Measure ? 


2 pints (pts.) 
8 quarts 
4 pecks 
36 bushels 


1 quart. 
1 peck. 
i bushel. 
1 chaldron. 


qt. 
pk. 
bus. 
ch. 



I -i gills (gi.) 
i 2 pints 
I 4 quarts 
i 63 gallons 
j 2 hogsheads 
I 2 pipes 
i 311 gallons 
i 42 gallons 
84 gallons 



1 
1 
1 
1 
1 
1 
1 
1 
1 puncheon, pun. 



pint. 

quart. 

gallon. 

hogshead. 

pipe. 

tun. 

barrel. 

tierce. 



pt. 
qt. 
gal. 
hhd. 

P- 
T. 

bbl. 
tier. 



What are the denominations of 
Cloth Measure ? 



21 inches (in.] 
4 nails 

4 quarters 
3 quarters 

5 quarters 

6 quarters 



1 nail. na. 

1 quarter. qr. 
1 yard. yd. 

1 ell Flemish. E.Fl. 
1 ellEnglish.E.E. 
1 ell French. E.Fr. 



What are the denominations of 
Circular Measure ? 



60 seconds (^') 

60 niinutes 

30 degrees 

90 degrees 

12 signs, or 360°, 



1 minute. ' 
1 degree. ° 
1 sign. S. 
1 quadrant. 
1 circle. C. 



What are the denominations of 
Long Measure ? 

3 barley-corns (b.c.) 1 inch. in. 
12 inches 1 foot. ft. 

3 feet 1 yard. yd. 

5J yds, 1 rod, pole, or perch, rd.p. 
40 rods 1 furlong, fur. 

8 furlongs 1 mile. M. 

60 geographic or 69|- statute 
miles, are one league. 

360 degrees, the circumference 
of the earth. 



44 



PRACTICAL ARITHMETIC. 



What are tlie denominations of 
Land or Square Measure ? 
144 square inches 1 sq. foot. ft. 
9 square feet 1 sq. yard. yd. 
30J sq. yards, or 

2721 sq. feet, 1 sq. rod. rd. 
40 square rods 1 sq. rood. r. 
4 square roods 1 sq. acre. A. 
640 square acres 1 sq. mile. M. 

What make a dozen? 
12 single things 1 dozen, doz- 
12 dozen 1 gross, gro. 

What make a score ? 
20 single things 1 score. 



5 score 



1 hundred. 



What are the denominations of 


Time? 






60 seconds 


1 minute 


. min. 


60 minutes 


1 hour. 


ho. 


24 hours 


Iday. 


da. 


7 days 


1 week. 


wk. 


4 weeks 


1 month. 


mo. 


12 months, or 365J 




days, 


1 year. 


yr. 


13 lunar months 


1 " 





What are the denominations of 
Solid or Cubic Measure ? 
1728 solid inches 1 sohd foot. ft. 
27 solid feet 1 solid yard. yd. 
40 feet of round 
timber, or 50 
feet of hewn 

timber, 1 ton. T. 

128 solid feet 1 cord wood. cd. 
Note. — A pile of wood 8 feet 
long, 4 wide, and 4 high, makes a 
cord, because 8 X 4 X 4 is 128. 

What make a quire ? 
24 sheets 1 quire, qr. 

20 quires 1 ream. rm. 

What make a hand? 
4 inches make 1 hand. 

What make a fathom ? 
6 feet make 1 fathom. 

What are the denominations of 
Ale or Beer Measure ? 



2 pints 
4 quarts 
36 gallons 
64 gallons 



1 quart. 
1 gallon. 
1 barrel. 
1 hogshead. 



qt. 
gal. 
bbl. 
hhd. 



The teacher should, as often as once a week, interrogate the whole 
school, or as many as are prepared for it, on these tables ; as it will 
be the means of reviving the pupil in their ready application. Many 
students^ after they have once memorized them, will in a short time 
be unable to repeat them ; in fact, we have often found it the case 
with many who have nearly, if not quite, gone through the arith- 
metic^ unable to repeat the different rules and denominations ; and 
for this reason they are placed in order and system for frequent refer- 
ence and recapitulation. 

Teacher. What is Circular motion used for ? 

Student. Circular motion is used to determine the motion and dis- 
tances of the planets, for the surveying of land, and to ascertain 
the course of a ship at sea. 

T. What is Time Measure used for ? 

S. Time Measure is very important ; by it we fix and establish 
dates of every kind. It regulates everything done or performed as 
to length of time. j 



PJRA C T I C A L_ A R I T H * E TIC. 45 

T. What is Solid or Cubic Measure used for ? 

S. Solid or Cubic Measure is used to determine the contents of 
solid bodies of any shape. 

T. What is Land or Square Measure used for ? 

S. Land or Square Pleasure is used bj surveyors in measuring and 
computing the quantity of land. 
T. What is Long Measure used for ? 

S. Long Measure is used to measure the distance of places, either 
land or water. 

T. What makes the difference between Geocjraphkal 3Ieasurea,nd Statute Measure f 

S. The difference is caused by the unevenness of the earth's sur- 
face. The Statute Mile is measured on the earth's surface. The 
Geographic mile is measured on a straight line through the air, or a 
bee-line, as it were, while the Statute runs up and down with the hills 
and mountains. 

T. What is the difference ? 

S. The difference is computed to be about 9i miles in a degree. 

T. What is the use of Wine Measure ? 

S, Wine Measure is to determine the quantity of wine, spirits, 
cider, vinegar, oil, mead, honey, kc. 

T. What is the use of Dry Measure? 

S. This measure is applied to all kinds of grain, seeds, roots, salt, 
sand, coal, oysters, &c. 

T. What is the use of Cloth Measure ? 

S. This measure is applied in buying and selling all kinds of cloths, 
ribbons, laces, &c. 

T. What is the use of Apothecaries' Weight ? 

aS'. This measure is used by druggists and physicians in mixing 
their medicines, though their articles are bought and sold by Avoir- 
dupois weight. 

T. What is the n?e of Avoirdupois Weight ? 

aS'. This measure is applied to all coarse articles and groceries, such 
as sugars, coffees, butter, cheese, meat, hay, hemp, fish, potash, and 
all the metals, except gold and silver. 

T. What is the use of Troy Weight ? 

S. Troy Weight is applied to the finer metals, as silver, gold, jewels, 
&c. 

T. By whom is Sterling Money computed? 

S. Sterling Money is the mode of reckoning money in Great Britain. 

T. By whom is Federal Money the mode of reckoning ? 

S. Federal Money» is the mode of reckoning money in the United 
States of America. 

Note. — The uses of the different measures should be often interrogated by the teacher. 
as well as the tables, &c. The teacher or learner writes out questions under each rule 
and example for solution. 



46 



PRACTICAL ABITHMETIC. 



2 

4 

5 
10 
20 
25 
33J 
50 



cents is 

a 

a 

a 
a 



50 



TABLE OF FRACTIONS 

of a dollar. 



2 cents is ^ of fifty cents. 

5 " A of " 
10 " iof 
25 '^ A of " 



$i is 25 cents. 
$1 is 50 " 
$1 is 75 " 



61 is- 
121 is- 
18i is- 
25 is- 
311 is- 
371 is- 
43| is- 
50 is- 
561 is- 
621 is- 
68f is- 
75 is- 
81-1 is- 
871 is- 
93f is- 



— lof |1. 

^ of $1. 

1 of " 



t. of $1. 
— f of $1. 

A of $1. 

1 of $1. 

» of $1. 



-11- of 

16 '^'■ 



of $1. 



J of $1. 

-M of $1. 
of $1. 



-if of $1. 



Teacher. Wliicli is the most important and practical rule in arithmetic ? 

Student. Proportion or Single Rule of Three. 

^. Why most important and practical ? 

S, Because questions are solved by their relative proportions. 

T. Why called Proportion or the Rule of Three ? 

S. Because there are three terms given to find a fourth. 

T. What other rules are pi'operly solved by Proportion ? 

S. Proportion is applicable in the solution of questions in Double 
Rule of Three, Practice, Interest, Insurance, Commission, Brokerage, 
Discount, Barter, Loss and G-ain, and all ijra^ical questions. 

T. How many terras in Proportion or Single Rule of Three ? 

S. There are three terms in the Single Rule of Three. 

T. Which of the terms must be placed in the third term ? 

S. Place that term in the third place which is of the same name or 
kind with that in which the answer is sought. 

T. Which of the two remaining terms will you place in the second term ? 

S. If the answer is to be greater than the third term, put the 
greater of the two remaining terms in the second term. 



PRACTICAL A R I T H JI E T I C. 47 

T. If the answer is to be less than the third term, which do you place in the second 
term? 

S. If the answer is to be less than the third term, put the less of 
the two remaining terms in the second place. 

T. What term will be the first term ? 

8. The first term is the one left after the second and third are 
placed. 

T. If more than one denomination are in the terms, how do you proceed? 

aS'. Bring the first and second terms to the lowest denomination 
mentioned in either, and the third term to the lowest denomination 
mentioned in it. 

T. Which of the terms do you multiply together ? 

8. I multiply the second and third terms together. 

T. By which of the terms do you divide that product ? 

S. I divide by the first term. 

T. Of what denoniination will that product be ? 

S. Of the same denomination of the third term. 

T. What must be done with the product or answer ? 

S. Reduce it to the same denomination in which the answer is re- 
quired. (That is, if necessary.) 

Example in the Single Rule of Three ; also applied in Practice, 
Interest, Insurance, Commission, Brokerage, Barter, Loss and Grain, 
&c. 

If 1ft) of cotton cost 7 cents, what will 360ft)s cost ? 

Now, you at once can determine that 3601]bs will cost more than 
lib : therefore the 360ft)s must be placed in the second place. The 
term to be put in the second place is the only difiiculty in solving 
practical questions. The third term is known by the answer, as it is 
to be of the same denomination. The first term is generally the con- 
sideration for the third terra, and the third is the answer for the first, 
term : hence the second term is the unknown term, and it is the con- 
sideration of the fourth term, and the fourth term is the answer 
for the second term. Hence the statement is, as lib : is to 360ibs : : 
so is 7 cents to both. This statement is preferable on account of 
convenience, as you multiply the second term by the tliird and divide 
by the first. The statement and operation will be in good form for 
the operation. 



48 



PRACTICAL AKITHMETIC. 



tb 3Jbs cts. 
1 : 360 : : 7 

7 



1)2520 



$25.20 Answer. 



360 

7 



2520 

or 

1 : 360 : : 7 

7 

1)2520 

25.20 
This shows the opera- 
tion on the blackboard 
to meet the changes. 




The result by i^ractice is 

the same. 

5 is'gLl 360 at 7 cts. 

1 is I'lS.OO at 5 cts. 
\\ 3.60 at 1 ct. 



lis 



3.60 at 1 ct. 



Ans. S25.20 at 7 cts. 



Interest. 
)360 at 7 per cent. 



Insurance. 
What is the insurance on $360 
7 worth of property, at 7 per 

cent. ? 

$25.20 Ans. (reckoned as 1 year.) $360 at 7 per cent. 
Commission. 7 

What is the Commission of $360 



at 7 per cent. ? 

$360 at 7 per cent. 

7 



$25.20 

Brokerage. 
What is the brokerage of $360 at 
7 per cent. ? 

$360 at 7 per cent. 
7 



$25.20 Answer. 

Barter. 
What is the difference in the barter 
of 360ft)s sugar and 360ft)S to- 
bacco ? The sugar to allow the 
tobacco 7 cts. per Hb. 
360ft)s 
7 



$25.20 

Loss and Gain. 
What is the loss on 360 yards of 
casimere at 7 cents per yard ? 
360 yds. at 7 cents. 
7 

$25.20 Ans. Loss. 



$25.20 sugar pays tobacco. 

Cain. 

What is the gain on 360 yards lace 

at 7 cents per yard ? 

Yds. 

360 

7 



$25.20 Gain. 



The amount is proven to be the same in Practice^ Interest^ 
Insurance^ Brokerage, Barter, Loss and Cain, at once showing the 
difficulty is in the construction or bearing of the question, and not in 



PRACTICAL ARITHMETIC. 49 

the operation ; and the teacher can do more real 360 

benefit in this way in one evening by changing the 7 

construction of the questions propounded, than he 

can accomplish in a week or month by the book or $25.20 Ans. 
usual mode. He may proceed in this way : What 
is the value of 360ft)s of cotton at 7 cts. per ft) ? 

What would be the interest of §360 at 7 per cent. ? $25.20. 

What would be the insurance of $360 at 7 per cent. ? $25.20. 

What is the commission of §360 at 7 per cent. ? $25.20. 

What is the difi'erence of $360 at 7 cts. diiference ? $25.20. 

What is the brokerage of $360 at 7 per cent. ? $25.20. 

What is the loss of 360 yds. at 7 cts. per yard ? $25.20. 

What is the gain on 360 yds. at 7 cts. per yard ? $25.20. 

You perceive the answer is the same in each example, and we have 
expressed the answer to each question, although the teacher need only 
take the 360ft)s cotton, or what amount he pleases, at 7 cents per lb, 
or at any other price, and work it out on the blackboard and the 
pupils to work the same on their slates, the teacher changing the 
construction or placing it as it were under a different rule, using the 
same quantity and valuation^ and it will exhibit the true answer to 
each question without ever changing the ^rst figure, &c. The teacher 
can still continue the process by varying the denomination to an end- 
less series of questions. And here we beg to remark that we have 
witnessed the proficiency of this system far surpassing that of the 
usual mode. This system, by an efficient and industrious teacher, 
will efiect more in a short time for practical calculation than the book 
or usual mode will effect in years, if during any time whatever. 

The same question further extended. 

What will 360ft)s cotton be worth at 7 cts. per tb ? $25.20. 

What will 360ft)s pork be worth at 7 cts. per ft) ? $25.20. 

What will 360 acres of land be worth at 7 dollars per acre ? 

$2520. 

What will 360 yds. of lace be worth at 7 pence per yd. ? 

2520 pence. 

If 1 cheese weigh 7 ft», what will 360 cheeses weigh ? 2520ft)s. 

If 1 horse can draw 7 bales cotton, what can 360 draw ? 

2520 bales. 

If 1 sheep cost 7 dollars, what will 360 cost ? $2520. 

If 1 yd. gold lace cost £7, what will 360 yards cost ? £2520. 

If 4 men reap 7 acres in one day, how many acres will they reap 
in 360 days ? 2520 acres. 

If 50 horses eat 7 bushel oats per day, how many bushels will they 
eat in 360 ? ' 2520 bus. 

If you travel 7 miles per hour, how far will you travel in 360 
hours ? 2520 M. 



50 



PRACTICAL ARITHMETIC. 



Should a hare leap 7 feet per leap, how many feet will he leap in 
360 leaps ? 2520 feet. 

If 1 week has 7 days, how many days will 360 weeks be ? 

2520 days. 
If 1 acre of ground produce 7 bus., how much will 360 produce ? 

2520 bus. 
What will 360 pmts of peas be worth at 7 cts. per pt. ? $25.20. 
What will 360 papers of needles be worth at 7 cts. per paper ? 

$25.20. 
If 1 hhd. sugar weighs 7 cwt., what will 360 hhds. 



weigh ? 
2520 cwt. 
2520 yds. 



How many yards in 360 pieces tape, 7 yds. each ? 

And so on, at pleasure. 

The teacher can extend the question at pleasure, including terms 
in all the denominations, and he will use his best efforts to impress 
and explain those exercises to the best advantage. This exercise is 
perhaps the best on construction that has been presented, and, if 
diligently applied, will render in a few months more practical bene- 
fit than years of the ordinary course. Children at a very early age 
will conceive application on this system. 

It IS not necessary to make but one operation of these questions : 
merely change the construction and denomination of the first one 
to meet each subsequent one. 



ii 



1. What will 561 bushels wheat be worth at 93f cts. per bushel ? 
hu. bus. cts. bu. bus. cts. 

1 : 561 ''^H or 1 : 56i : : 93| 
4 4 4 4 4 4 



Now, 



4 225 375 

Fourths. Fourths. Fourths. 

4 : 225 : : 375 

375 



4 225 
4 ^5 

16 1125 
1575 

675 



375 



1125 
1575 
675 

4)84375 + 



4)210931- 
$52.73^ 



16)84375(52.73^t 
80 



117 
112 



The second and 
third terms are 
brought to J : 
therefore their pro- 
duct 84375 is 
16ths, the divisor: 
hence 4ths multi- 
plied by 4ths 
renders their pro- 
duct 16ths. 



55 
48 



PRACTICAL ARITHMETIC. 51 

2. If 1 yd. of cloth cost $1|, what will ISJ yds. cost ? 

yd. yds. $ 
1 : 18f : : If 
4 _4 _8 

4-75 - 11 
8 11 

Now, the second term 18f is brought to 4ths, and the 

32)825(25.78i third term If to 8ths, and by multiplying 4ths by 8ths 

54 their product is 32ds, that is, the 825 is 825 thirty- 

twos : hence thirty-two is the divisor, giving the answer 

185 $25 and 25 over, which is 25-32ds, Add two ciphers 

160 to obtain the cents, which gives 78 cents and 4 remain- 

~^rh ing, which is 4-32ds, and 4-32ds reduced gives l-8th. 

224 

260 
256 

8. When 1 gallon of syrup costs 33J cents, what will 68f gallons 
cost ? 

gal. gals. cts. 
1 : 68| : : 33J ^ 

4 4 o Now, the third term is reduced to thirds, the second 

4 275 —Too to 4ths, (fourths:) consequently their products are 12ths, 

Q -if\(\ ^^ fourths multiplied by thirds produce twelfths, the di- 



visor. 



12) 27500 



$22.91| Ans. 

Teacher. In the first example, what is the true denomination of 84375 ? 

Student. The 84375 is 84375-16ths. 

T. What is the 825 reckoned in No. 2 ? 

S. The 825 is 825-32ds. 

T. In No. 3 is the 27500 reckoned whole numbers ? 

S. They are not, as two terms consisting of fourths and thirds 
have been multiplied together, their product is twelfths. 

Note. — Where fractional numbers are multiplied, to reduce them to whole numbers 
you must divide by the product of fractions multiplied, as foutrhs by thirds render 
twelfths. 



52 PRACTICAL ARITHMETIC. 

4. How much will 9| yds. broadcloth amount to at $3| per yd. ? 

yd, yds. ? 

1 : 9| : : 3J 5. If 1 yd. of ribbon cost 2J cents, 
8 8 8 what will 21} yds. cost ? 

8-77 -~3i Eiffhths. . «/^- yds. cts. 

8 31 l:21f::2| 



64 77 



4 4 2 



231 4-87 - 5 

64)2387[sixty-fourths](37.29^ Answer. ? ^ 

467 54f cents, Ans. 

448 In No. 4, eighths are multiplied, which 

■^p,^ / J J x" A \ makes their product 64ths (sixty -fourths). 

lyU (ada tor cents.) in Nq. 5, fourths and 2s or halves are multi- 

128 plied, which renders their product eighths; 

nc)f\ and so it is with any two fraction numbers. 

"-"^ Fourths by fourths renders their product 

576 sixteenths, 2 by 2s or J by ^fourths, Sds by 

'^Ma(iX ^*^^ ^^*^^' ^^^^ ^y ^*^s 25ths, 8ths by 

?/6fVi6 8ths 64ths, and so on, with 'any fractional 
numbers. 



PROPORTION OR RULE OF THREE IN FRACTIONS. 

The rule of three in Fractions generally consists in mixed numbers 
in all their parts, and the statement is the same as in the single rule 
of three. 

Teacher. How do yon state a question in the single rule in Fractions ? 

Student. The same as with the single rule. 

T. Suppose all the terms are mixed numbers, how will you proceed ? 
S. I will bring all the terms to improper fractions. 
T. Then how will you proceed ? 

S, I will invert the first or dividing term, and multiply all the top 
terms together, and all the under terms together. (This includes the 
rule.) 

T. What are the top terms called ? 

S, The top terms are denominated the numerator."^ 

T. What are the under terms denominated ? 

S. The under terms are denominated the denominator ;* as, of JJ 17 
is the numerator and 18 the denominator. 

* Refer to Dictionary in regard of words. 



PRACTICAL ARITHMETIC. 53 

1. If f of a yard of ribbon cost 18| cents, what will 27f yards 
cost ? 1 

yds, yds. cts. 
Invert f : 27f : : 18f 
4 4 

111 76 Now, the f is the first term, which is to be 

, , * ^ inverted. The second and third terms are 

yds. yds. ' brought to fourths; then 8, 111, and 75 are 

f X ^1 X ^x^ = ««A°^ multiplied together, making 66600, and the 

4 5_, 4, and 4, which make 80, and 66600 di- 
8n\fifiRnn vided by the 80 gives the answer, $8.32^. 

16 

5 $8.32J 

80 

2. How many yards of carpeting that is If yards wide will be re- 
quired to cover a floor that is 4 J yards long and 3| yards wide ? 

yds. yds. yds. To reduce a fraction to its lowest terms, as in this 

If : 3| : : 4i- example, the 96 is a remainder, which is ^ ; and 3 

4 5 3 will divide either without a remainder. 

— 105)936(811 yards. 

Invert I i/ i/ 840 

yds. yds. yds. 



I X 1/ X ^-i = Mf 3) 96 (32 

^ 3)105(35 

9 
l5 _ 

7 15 



105 



15 



3. If f of a yard of lace cost f of a cent, what will 33f yards 
cost ? 

f : f : : 33f 48)1620(33f cents. 

4 144 



i|5 180 

I X f X If = i|2o 144 

4 • 

— 12)36(3 

If 12)48(4 



48 



54 PRACTICAL ARITHMETIC. 

4. If I of a pound of candy cost 3f cents, what will 20 pounds cost ? 

invert s ' ^ ' • o-g It does not necessarily require all the 

5 terms to be of mixed numbers, as in Ex- 

ample No. 4 the second term is a whole 

y number, (20,) which is ^^. 

I X \« X V = 'H' 25)2720($1.08t Answer. 

5 25 

5 220 

5 200 

25 5)20(4 

/ 5)25(5 

DOUBLE RULE OF THREE. 

Teacher. What is the Double Rule of Three ? 

Student. The Double Rule of Three is that in which five terms are 
given to find the sixth. 

T. What is the nature of the terms ? 

5. Three of the terms are a supposition and two a demand. 

T. Which is the first term to be considered in a statement in the Double Rule of 
Three ? 

aS^. We are to consider which of the terms is of the same kind with 
that in which the answer is required, and set that term in the third 
place. 

T. How many terms will there be left in the question ? 

S. A question in the Double Rule has two first terms, and two 
second terms or double terms. 

T. How do you state those double terms ? 

S. We must consider each pair of similar terms separately, and the 
third one as the terms of a statement in the Single Rule, and, if the 
answer is to be greater than the third term, place the greater in the 
second place and the less in the first, and so with the two remaining 
terms ; if the answer is to be greater than the third, place the larger 
also in the second place and the less in the first. 

T. Should the answer be less than the third term, how will you proceed ? 

S. Reverse the first and second terms, placing the less in the 
second term and the greater iij the first. 

T. Should the nature of the question require the less in one and greater in the 
other, how will you proceed ? 

S. I will then place one of the greater and one of the less terms 
in the second and first terms, according to the sense of construction. 

T. After you have made your statement, how will you proceed ? 

S. I will multiply the two first terms together for a divisor, the two 
second terms together, and their product by the third. 



PRACTICAL ARITHMETIC. 



55 



1. If 72 pounds of bread be sufficient for IB men 20 days, how 
many pounds will be required for 48 men 30 days ? 
Men, 18 : 48 
Days, 20 : 30 



}- 



72ibs. 



360 1440 

72 



2880 
10080 

360)103680(2881)s, Ans. 
720 



3168 
2880 



NoWjfromthe sense of the questian, 
the answer is required in pounds: 
therefore, 72 lbs is put in the third 
tei'm. And we know 48 men will eat 
more in the same time than 18 : there- 
fore the 48 is placed in the second 
term and 18 in the first term. Also, 
we know that it Avill require more 
pounds for 30 days than for 20: there- 
fore the 30 is placed in the second 
and the 20 in the first, as the rule 
directs. 



2880 
2880 

T. Can you solve a question in the Double Rule of Three by any other method ? 

*S'. By two statements in the Single Rule of Three we can solve 
questions in the Double Rule. 

Example in the above question by two statements. 

ft)S. 



men. men. 
18 : 48 : 

72 

96 
336 



72 



Now, if 18 men in a certain time eat 72 lbs of 
bread, 48 men will eat 192 lbs in the same time; 
and, if 20 days consume 192 lbs, 30 days will 
consume 288 lbs, Ans. 



18)3456(192 lbs. 
18 

165 
162 

36 
36 



da. da. 

20: 30 

192 

60 
270 
30 



Sbs. 
192 



2,0)576,0 



288 ft)s. Answer. 



2. If 28 men in 20 days earn $170, in what time will 8 men earn 
$34 ? 



56 PRACTICAL ARITHMETIC. 

Men, 8 : 28 



170:' 34 r ^=20 days. 



560 112 

8 84 



1360 952 
20 



1360)19040(14 days. 
1360 



This question requires a different 
statement to tlie preceding one ; for 
one of the conditions requires the 
answer to be greater than the third 
term, and the other condition re- 
qiiires it to be less : therefore one of 
the larger terms is placed in the 
second term and one of the less. 



5440 
5440 



By two statements in the Single Rule. 
men. men. days. % % days. 
8 : 28 : : 20 170 : 34 : : 70 
20 70 



8)560 



Now, it will take 8 men 

longer to perform a piece 

of work than 28 men : so 

the first answer must be 

greater than the third 

170)2380(14 days, Ans. *f"^- Jl^e answer is 70 

' „ ^ •^ ^ days, the time it would 

i ( U require 8 men to earn the 

70 days. $170, while the 28 men 

Require 8 men 70 days to do the 680 earned it in 20 days. Now, 

work of 28 men for 20 days. 680 Vlr^T.^:: i^l 
I put the larger term in the first and $34 in the second place, as it would require less 
time to earn $34 than $170. 



3. If $200 gain $12 in 24 months, how much would $1000 gain 
in 8 months ? 

$200 : 1000 



24 



} : : $12 



800 8000 
400 12 



4800)96000(20$ 
9600 



It is customary, when the dividend 
and divisor have ciphers, to cut off as 
many ciphers of the divisor as the 
dividend has, although for fulness it 
has been omitted in those examples. 



I 

4. If $100 gain $6 in 12 months, what amount will gain $10 in 4 

months ? 



practical arithmetic. 57 
m!4:;o}::§100 

• J Now, when the nature of a question does not 

24 1-0 ^^ sense present the idea that the first term in 

-j r\r\ Single Rule of Three is the consideration of the 

third term, and the third the answer to the first, 

24)12000(500$ it is not direct proportion, but is called Inverse 

1 on Proportion. 



00 



Example. 



If 12 men can do a piece of work in 36 days, how long will it re- 
quire 24 to do it ? 

men. men. days. 
24 : 12 : : 36 24)432(18 Ans. 

36 24^ 

72 192 



192 



432 



PRACTICE. 



Teacher. What is Practice ? 

Student. Practice is an abbreviation of the Single Rule of Three 
in such questions as have 1 for their first term. 

T. When the given price is J^, ^, |-, |^, f, &;c., how do you proceed? 

*S^. Divide the given number by the aliquot parts of a whole num- 
ber of the same denomination. 

T. When the given price is a whole nui^iber, how do you proceed ? 

S. Divide the given quantity by the next higher denomination for 
the answer in that denomination. 

T. When the given price is of various denominations, how do you proceed ? 

S. Set down the given price of one of the highest given denomina- 
tions and multiply it by the highest denomination, and take aliquot 
parts of the remaining denominations. 

T. After you have multiplied by the highest denomination and taken aliquot parts 
for the remaining denominatiou.s, hov/ do yuu proceed ? 

*S^. Add their products together for the answer. 

T. How do you explain question 3d ? 

;S^. If the given price is mills, divide by the aliquot parts of a cent ; 
if cents, divide by the aliquot parts of $1 ; if farthings, by the aliquot 
parts of a penny ; if pence, by the aliquot parts of a shilling ; if 
shillings, by the aliquot parts of a £ ; if qrs., by the aliquot parts of 
a cwt. ; if ft)s., by the aliquot parts of a qr. ; if pecks, by the aliquot 
parts of a bushel ; if quarts, by the aliquot parts of a pk. ; and so on, 
with all the denominations, taking aliquot parts of a lesser denomina- 
tion of a higher, kc. 



58 



PEACTICAL ARITHMETIC. 



EXAMPLES UNDER QUESTION SECOND. 



I cts. 



296 yards (c 

No.] 

296 



f IS 

i is 



h 



148 
74 



at 
at 



the same 



J or 50 cts. is 
i or 25 cts. is 



j dollars. 

No. 2. 
296 

T48 at 
74 at 



^ or 50 cts. 
i or 25 cts. 



.22Ans. f 



$222Ans.|f or 75 cts. 



Now, in the first example, if you multiply 296 by 5 mills, which is ^ cent, you have 
$1.48 ; and multiply $1.48 by 5 mills, which is ^, you have 74 cents. And the same is 
true of No. 2. Multiply 296 by 50 cts., which is one half-dollar, and you have $148, 
and then the 296 by 25 cts., and you have the $74. 



No. 3. No. 4. 

296 yards @ ^^, the same 
3^1^1296(18.50 i|296 



No. 5. 



16 

136 

128 



$37 



8.00 (for cents.) 
80 



50 

25 




No. 6, the same 
296 



is 


i 


is 


1 

'A 


IS 


i 



148 at 50 cents. 
74 at 25 
37 at 12i " 



at 871 or I 



1 the 


same 


@lf 


50 


is 
is 


i 


296 


25 


148 


121 


is 


1 


74 


H 


is 


i 


37 



Money) and to business men than any 
Practice. 



at 50 cents. 

at 25 '' 

at 12| '' 

18.50 at 61 " 

$277.50 at 93f or If 

Now, by multiplying No. 3 by 6|^, No. 
4 by 121 No. 5 by 93f, No. 6 by 87^, 
you have the respective answers to 
each as produced by the aliquot parts. 
And in No. 5 and 6, multiplying by 50, 
25, 12J, and 6|-, you obtain the same 
result as produced by the aliquot parts 
as indicated in the examples. These 
six examples are of more importance 
to us (Americans reckoning by Federal 
other features in the system as obtained by 



7. What is the value of 2267 yards (or any thing) at 95 cents ? 



50 is 


h 


25 is 


^ 


10 is 


^ 


10 is 


I 



2267 



1133.50 at 50 cents. 
566.75 at 25 '' 
226.70 at 10 " 
226.70 at 10 " 



$2153.65 at 95 



In No. 7 you see that 10 is not the 
l-5th of 25, but the l-5th of 50 : con- 
sequently each fifth is taken from pro- 
duct of 50 or J, being 1133.50, and 
l-5th is 226.70 for each l-5th. 



PRACTICAL ARITHMETIC. 



59 



TABLE OF POUNDS, SHILLINGS, AND PENCE. 



2 Shillings is ^V of a £. 

4 Shillings is | " 

5 Shillings is i " 
10 Shillings is i " 



2 Pence is ^ of a Shilling. 

3 Pence is J " 

4 Pence is J " 



6 Pence is J 



8. What is the value of 888 (any things) @ 10 pence ? 
6 is 'J- 888 

3 is ;J 444 shillings at 6 pence. 

1 is !j|222 shillings at 3 pence. 

74 shillings at 1 pence. 

20)740 shillings at 10 pence. 

<£37 

Operation 



Now, 888 by 10 pence would 
make 8880 pence ; and that reduced 
makes 740 shillings, or £37. So 
the 888 at 6 pence will make 444 
shillings, at 3 pence, 222, at 1 pence, 
74, and their product reduced 
makes £87. 



888 at 10 pence. 
10 


888 at 6 pence. 
6 


12)8880 pence. 


12)5328 pence. 


20)740 shillings. 


444 shillings. 


«£37 


888 at 3 pence. 
3 


888 at 1 pence. 


12)2664 pence. 


1 

12)888 

74 shillings. 


222 shillings at 3 pence. 
444 " 6 " 

74 ^' 1 " 




20"J740 shillings. 




£37 



9. What is the value of 1286 @ 19 shillings ? 



10 is 

5 is 
*4 is 



1286 



643 




at 10 shillings. 


321 10 


at 


5 


u 


257 


4 


at 


4 


il 



£1221 14s. @ 19s. 



Now, 4* is not the l-5th of 
5 nor of 10, but of 20, £1 : 
consequently the ^j and l-5th 
are taken out of the given 
number, (1236.) 



60 



10. 



11. 



IS 



4 is 



What 
10 is 


is 

-J 

is 
s. 

7 


PRACTICAL ARITHMETIC. 

the value of 12860 @ £2 15s. 9(^. ? 
12860 

2 


5 is 

*6 is 

3 is 

What 
£ 
I 3 


25720 at £2 
6430 at 10 shillings. 
3215 at 5 

15 

321 10 at 6 pence. 
160 15 at 3 pence. 

£35847 5s. 9d, 

the value of 2 cwt. 1 qr. 

d. 12. Wha 

6 2 qts. 

2 2 is 


In No. 10, 6* is not tlie 1-lOth 
of 5, but the 1-lOth of 5 shil- 
lings, which is 60 pence : hence 
6 pence is the 1-lOth of 60 pence 
or of 5 shillings. 

4 ») at £3 7s. 6d. ? 
t is the value of 120 bu. 2 pks. 
at 45 cents ? 
1 120 bus. 
45 

600 
480 

1 
1 5400 at 45 

22.5 for 2 pecks. 
2.8i for 4 quarts. 


^ 6 15 for 2 cwt. 
16 lOJ for 1 qr. 
2 4f for 4 ft)s. 


£7 1- 


k. 


Sid. 2 is 



$54.25.3J 
Any question in Practice can be solved and proven by the Single 
Rule. You can solve any question in Practice on the same principles 
as the examples given, although all the denominations are not pre- 
sented, nor is it necessary, as all are solved on the same principle, 
according to the denomination. 



TARE AND TRET. 



Teacher. What are Tare and Tret ? 

Student Tare and Tret are allowances made by merchants on goods 
thftat are bought and sold. 
T. What is the nature of Tare ? 

S. Tare is the weight of the box, hogshead, barrel, bag, or what- 
ever contains the goods sold. 

T. What is Tret ? 

*S'. Tret is allowance for waste, dust, or whatever may affect the 
weight. 

T. What is Gross? 

S. Gross is the weight of the goods sold, including box, barrel, bag, 
and whatever affects the weight of the same. 



\ TRACTICAL ARITHMETIC. 61 

T. What is theYemainder called after the tai'e has been deducted ? 

S, After the tare is deducted, the remainder is called suttle. 

T. What is the remaindei- called after all allowances have been made ? 

iS*. After all allowances have been deducted, the remainder is the 
nett weight of the goods sold. 

When the tare is so much on the box, bag, barrel, you are to sub- 
tract the tare from the given quantity. When the given tare is so 
much per box, barrel, bag, &c., multiply the number of bags, boxes, 
barrels, (fee, by the given tare, and subtract the product from the 
given quantity. When the tare is so much on the hundred-weight, 
divide the gross weight by the aliquot part or parts of a hundred- 
weight and deduct the amount of the result from the gross, and the 
remainder will be the nett. When Tare and Tret are both allowed, 
first subtract the Tare and then the Tret, and the remainder will be 
the neat. 

Example. 

1. What is the nett"^ of 20 hhds. of sugar, each weighing 1080 ibs, 
an allowance on each hogshead for tare of 90 ibs each, and the worth 
at 7} cents per ft) ? 

Operation. 

19800 ft)s @ 7f . 
20 the number. Tare, 90 each. 7 

20 the number. 

21600 ft)s for the twenty. ^ . 

1800 tare subtract. 1800 amt. of tare. ^ ^^ 



1080 on hhd. 1 



138600 
9900 for J. 
4950 for J. 



19800 ft)s. nett @ 7|- cts. $1534.50 Ans. 

2. What is the neat weight of 7 casks indigo, each weighing 3 cwt. 
2 qrs. 12 ft)s, tare 25 R)s per cask, also the price, at $1^ per ft) ? 

Opi'ration. 
cwt. qrs. ft)s. 25 ft)s per cask. 

3 2 12 each. 7 number. 
7 the number. 25)175(7 qrs. 4)7 



25- 1 - 
1 3 


- 9 gros 
off. 


23 - 2 - 
4 


- 9 neat 


94 qrs. 
25 




479 





175 1 cwt. 3 qr. 



1)2359 ftjs @ IJ 
li 
2359 at 1. 
589.75 at J. 



$2948.75 Ans. 



2359 ft3s. 



* Nett is sometimes written net, and fgniu neat. 
5 



62 



PRACTICAL ARITHMETIC. 



3. What is tlie neat weight and wortli of 20 boxes raisins, each 
gross 1 cwt. 1 qr. 18 ibs, tare 22 Bbs per cwt., at $5.50 per cwt. ? 

$5.50 



20 is 



^ IS 



cwt. qr. ibs. 

1 1 18 each. 

20 number. 



22 



1100 
1100 



28 
6 



10 for 20 boxes 
4 tare off. 



22 
cwt. 



1 

qr. 



22 
7 



4 tare. 

6 nett. 
ibs. 



121.00 for 22 cwt. 
1.70.5 for 1 qr. 6 



S122.70.5 



qr. 1 is 

Sbs. 5 is 
lis 



5.50 



1.37.5 for 1 qr. 

27.5 for 5 ibs. 

5.b for 1 ib. 



..70.5forlqr.6 



4. What is the neat measure of 30 hhds. of molasses, containing 
each 94 J gals., tret 7 gals, per hhd., and what is the value at 31 J 
cents per gallon ? 



94J gals. each. 
30 number of hhds. 



2820 
15 

2835 gals, gross measure. 
210 



7 gals. each. 
30 m. of hhds. 

210 gals. tret. 



gals. cts. 
2625 @ 311 



31i 



2625 
7875 



656i 



2625 gals. nett. 



$820,311 Ans. 



DUTIES. 

Teacher. What is duty ? 

Student. Duty is a tax imposed by the general government on 
merchandise brought from other countries. 
T. For what purpose is duty imposed by government ? 

S. Duty is imposed on foreign goods to raise money for govern- 
mental expenditures, &c. 

T. How many sorts of duties ? 

S. There are two kinds of duties, a Specific and Ad Valorem duty. 

T. What is Specific Duty ? 

S. Specific duty is so much imposed on the quantity, as for ton, 
cwt., gallon, yard, &c. 

T. What is Ad valorem duty ? 

S. Ad Valorem duty is so much imposed or levied on the cost of 
articles bought. 

T. How do you ascertain a Specific duty ? 

S. By finding the neat weight or quantity, as in Tare and Tret, 
and multiplying it by the given duty. 



PRACTICAL ARITHMETIC. 63 

T. How do 3'ou ascertain Ad Valorem duty ? 

S. By multiplying the cost by the duty imposed or levied as rate 
per cent. 

1. What is the specific duty on 20 boxes of St. Croix sugar, each 
weighing 270 Sbs, tare 6 lbs per box, tret 3 ibs per box, duty 1 J ? 

Operation. ^ 
270 ibs each. 20 boxes. 

20 boxes. 6 lbs tare each. 



5400 gross. 

120 tare off. 120 ft)s tare. 

5280 suttle. 

60 tret off. 20 boxes. 

Nett, 5220 lbs @ IJ. ^ ^^ t^et each. 

1? 

52:20~atl. 60ft)stret. 

13.05 at 1. 

?65.25 duty. 

2. A merchant imported 14 pieces broadcloth, 22 yards each, 
which was invoiced at $2J per yard. What is the duty ad valorem at 
15 per cent. ? 

Operation. 

14 pieces. $693 at 15 per cent. 

22 yards each. 15 

28 3465 

28 693 



308 yards. $103.95 duty. 

2J cost per yard. 

616 at $2. 
77 at 1 

$693 cost. 21$. 



BARTER. 

Teacher. What is Barter? 

Student. Barter is the exchanging of one thing or commodity for 
another, at such prices as are mutually agreed on. 

T. What must be known to make an equitable exchange in goods, &c. ? 

S. The quantity and price of one of the articles to be exchanged, 
must be known. 



64 PRACTICAL ARITHMETIC. 

T. If the quantity and price of one of the articles or commodity are known, how 
will you proceed to make the equitable exchange ? 

8. Bj multiplying the quantity and the price that is given to fin 
its value, which I will divide by the price of the other article. 



w 

1 



1. A has 6 bags of coffee, w^eighing 980 fbs, at 15 cents per ft), 
which he wishes to exchange vv^ith B for sugar at 9 cents per ft). How 
much sugar must A receive ? 

Operation. 
980 ft)s coffee at 15 cents. cts. % ft) 
15 9 : 147.00 : : 1 
1 



4900 



980 9)14700 



$147.00 worth of coffee.., IGSSJ ft)s sugar. 

Note, — In Barter, a statement in each commodity, as in Single Rule of Three, ac- 
cording to the sense, obtains the true result. Recollect, we have on a former occa- 
sion explained the applicatio]^ of the Single Rule to Barter and other rules, &c., which 
will be further extended in other rules. 

2. How much corn at 55 cents is equal to 630 ft)s sugar at llj- 
cents per ft) ? 

Operation. 
ft) ft)s. cts. cts. S bus. 

1 : 630 : : lU 55 : 72.45 : : 1 

11} -^ • 1 



69.30 at 11 cts. 55)7245(131i^bus. 
3.15 at 1 ct. bb 

S72.45 the value of sua^ar. ^^^ 55 cents is worth 

^ "95 $72.4-5. 



Thus, bb 

131t\ 8 suo^. 

55 bb ^^ 

'^bb 40 

<6bb . 40 

11)440 



Ti 



40 for i\- bus. 



$72.45 -"40 for 



Note. — In cases where fractions are connected, as in the last, — 131 8-1 1th bus. at 
55 cents, multiply the top or numerator by the price, and divide by the lower or 
denominator. 



PRACTICAL ARITHMETIC. 65 



EQUATION. 

Teacher. Foi' what purpose is Equation used ? 

Student. Equation is used to ascertain the mean or equated time 
of several payments, due at different times, without loss on either 
side. 

T. "What rule will you give for Equation of Payments ? 

S. Multiply each payment by its time, and divide their product by 
the sum of the payments, (which is the whole sum,) and the quotient 
will be the true time. 

1. X owes B $600, to be paid as follows : $150 in 5 months, $150 
in 7 months, and $300 in 9 months. But it is agreed to pay the whole 
amount at one time. When must it be paid ? 

Operation. 

$150 in 5 months, 150 x 5 = $750 
$150 in 7 months, 150 x 7 = $1050 
$300 in 9 months, 300 x 9 = $2700 



Sum, $600 $4500 product of sum and time. 
6,00)45,00 600)4500(71 months. 

4200 

7^ months, or 



M0'\300('l 
300>/600\2 



2. Suppose a merchant has due him $1200, $200 to be paid in 4 
months, §400 in 6 months, $600 in 8 months. When is the equated 
time for the Avhole amount ? 

Operation. 

$200 in 4 months, 200 X 4 = $800 1200)8000(6f months. 
$400 in 6 months, 400 X 6 = $2400 7200 
$600 in 8 months, 600 X 8 = $4800 

400^ 800 (2l 

~~ooU 



Sum, $1200 



?00yi200\ 



INTEREST. 



Teacher. What is Interest? 

Student. Interest is the sum which is paid for the use of borrowed 
money. 

T. Wliat four things must be taken into account in computing interest ? 

aS'. In computing Interest, principal, rate per cent., time, and 
amount must be observed. 



Q6 PRACTICALARITHMETIC. 

T. What is principal ? 

S. The sum for which interest is to be computed is principal. 

T. What is rate per cent. ? 

S. Rate per cent, is the sum to be paid for the use of $100 or 
^100 for one year, or so much per % or per £j. 

T. What is time? 

8. Time is the number of years, months,- and days for which 
interest is to be computed. 

T. What is amount ? 

8. Amount is the principal and interest added together. 

T. How do you find the interest for 1 year ? 

B, Multiply the amount by the rate per cent, and cut off two 
right-hand figures for cents : the rest will be dollars. 
T. Should the amount be dollars and cents, how do you proceed ? 

8. When the amount is dollars and cents, multiply by the rate per 
cent., cut off four right-hand figures : the first two will be for mills, the 
second will be ceiits, the remainder dollars. 

T. When the time is for two, three, four, &c. years, how do you obtain the interest ? 

8. Compute the interest for 1 year and then multiply that interest 
by the number of years to be computed. 

T. Suppose the rate per cent, was a fraction, or a whole number and a fraction, 
how do you proceed ? 

8, The aliquot part or parts of the principal #iust be taken for 
such fractional rates. 

T. How do you find the interest for months, or years and months ? 

8. Multiply the interest of 1 year by the number of years and 
take aliquot parts for the months ; or multiply the principal by half 
the number of months and cut off two right-hand figures for cents, 
which will give the interest at 6 per cent. 

T. How will you find the interest at any given rate per cent. ? 

8. Multiply the interest found at 6 per cent, by the given rate 
per cent, and divide by six. 

T. How will you find the interest for days, or months and days ? 

8. Find the interest for the months as before, and for the days multi- 
ply the principal by the number of days and divide by six, and the 
quotient will be the interest in mills at 6 per cent. ; and by cutting 
off three right-hand figures the remainder will be dollars, and the in- 
terest of the days added to the interest of the months will be the 
answer, or amount of interest. 

T. If at any rate per cent, other than six, how will you obtain the answer ? 

aS'. Multiply the principal by the number of days and their product 
by the given rate per cent, and divide by 36, and the quotient will 
be the answer in mills. 

T. How do you find the principal when the time, rate per cent., and amount are 
given ? 

8, I will find the amount of %\ at the given rate and time and 
divide the given amount by it. 

T. How will you ascertain the rate per cent, when the principal, time, and interest 
are given ? 



PRACTICAL ARITHMETIC. 67 

S. If the interest be dollars, I will annex two ciphers to it and 
multiply the time and principal and divide the interest, with two 
ciphers annexed, by it. 

T. How will you find the time when the principal, interest, and rate per cent, are 
given ? 

aS'. Annex two ciphers to the given interest if the interest be 
dollars only, and divide it by the product of the rate per cent, and 
principal. 

Note. — Now, in multiplying dollars by cents, of course their product is cents, 
which is fully taught under the change of construction in the Single Rule. And you 
are to bear in mind when dollars are multiplied by cents their product is cents, 
and if dollars and cents are multiplied by cents their product is less than mills. 
Where dollars only, you cut off two right-hand figures; but if dollars and cents, you cut 
off four right-hand figures. See another part of this work, page 106. 

To find the interest for one year or any, number of years. 
What is the interest of $5840 at 5, 6, and 7 per cent. ? 

Operation. 

No. 1. No. 2. No. 3. 

?5840 @ 5 per cent. §5840 @ 6 per cent. §5840 @ 7 per cent. 
5 6 7 



$292.00 for 1. $350.40 for 1. $408.80 for 1. 

2 5 7 



$584.00 for 2 years. $1752.00 for 5 years. $2861.60 for 7 years. 

Note. — Add the principal, 5840, to the interest in either case, and it will be the 
amount. As in No. 1, for 1 year the interest is $292 ; add the principal, $5840, and 
$6132 is the amount, and so on. 

What is the interest of $4385.50 at 7 per cent. ? 

$4385.50 

7 



You see, as this amount has dollars and 
Interest for 1 year, $306.98.50 cents, you have to cut off four right- 

3 hand figures. 



Interest for 3 years, $920.95.50 

To find the interest for months^ or years and months. 
What is the interest of $560 for 10 months at 5 per cent. ? 



as 



PRACTICAL ARITHMETIC. 



560 

5 



or 



i|28.00 for 1 year. 
1 

2 



14.00 for 6 months. 
7.00 for 3 months. 
2.33-J for 1 month. 

.331 for 10 months. 



Ope7'ation. 



560 

5 half the number of months. 



28.00 for 10 months at 6 per cent. 
5 rate. 



6) 14000 
$23,331 at 5. 



What is the interest of $980 for 2 years and 9 months at 7 per 
cent. ? 

Operation. 

2 years and 9 months = 33 months. 
2)33 



or 



m 



68.60 for 1 year. 

2 



16J half the number. 



137.20 for 2 years. 
34.30 for 6 months. 
17.15 for 3 months. 

$188^5 for 2 years 9 mos. 

By Rule of Three, $980 for 1 
year is 



$980 for 2 years and 9 months. 
16J- half the number of months. 



5880 
980 
490 



.60. 



mo. yrs. mo 
Then, if 12 : 2 9 : 
12 

~33 



: 68.60 

r> 


$161.70 at 6 per cent. 

7 


33 

20580 
20580 


6)113190 

$188.65 at 7 per cent. 



12) 226380 
$188765 



$980.80 for the same time and 
7 rate per cent. 



$980.80 



16-J half the months. 



68.65.60 for 1 year. 

2 



588480 
98080 
49040 



137.31.20 for 2 years. 
34.32.80 for 6 months. 
17.16.40 for 3 months. 



$188.80.40 



$161.83.20 at 6 per cent. 

7 the given per cent. 
6) 11328240 
$188.80.40 at 7 per cent. 



r R A C T I C A L ARITHMETIC. 69 

Or by the Single Rule, thus : mo. Y. mo. 

S ? cts. 12 : 2 9 : : 68.65.60 

1 : 980.80 : : 7 12 33 

7 33 mo. 2059680 

■; — r~ 2059680 

l)686o60^ 12)22656480 

68.^5.60 $188^40 



To fiiid the interest for days, or months and days. 
What is the interest of §560 for 15 days at 6 and 7 per cent. ? 

Operation. 

$560 at 6 per cent. $560 at 7 per cent. 

15 time. 15 

"2800 



2800 560 

560 



6)8400 



8400 

7 rate. 



1.40.0 for 15 days. 



36)58800(16331 Mills. " 
36 



Or by the Single Rule, 9-1 g 



560 at 6 per cent. 
6 



120 

108 



33.60 for 1 year. 120 

da. da. $ L^i 

Then, if 365 : 15 : : 33.60 12)M(J 

15 

. Now, if we divide or make the first term 

16800 360 days, the answer will be $1.40. 

o^.rnnTTnTv-. 00 « ^'^us, 360)50400(1.40 

365)50400(1.38^ ^360 ^ 
365 

1390 1440 

1095_ 1440 

2950 

^q;^^ Now, 30 days to the month give 360 days 

^^^^ to the year. 



^^-%(f^ 



5;^§^Vt3 



70 PRACTICAL ARITHMETIC. 

"What is the interest of §580 for 2 years 4 mos. 9 days at 6 per 
cent. ? 

580 at 6 per cent. 
14 months is half of 2 years and 4 months. 
2320 580 

580 9 days. 



Interest for 28 mos., $81.20 at 6 
For 9 days, 87 

$82.07 
The same at 7 per cent. 
580 
14 half the months. 


per 


cent. 6)5220 

87.0cts. for 9days. 

580 

9 the days. 
5220 


2320 
580 

81.20 at 6 per cent. 

7 
6)568.40 




7 th§ given per cent. 

36)36540(1.01.5 for 9 days. 
36 

54 
36 



$94,731 at 7 per cent, for 2 yrs. and 4 mos. 180 
1.01.5 at 7 per cent, for 9 days. 180 

$95.74.81 

Note. — The questions preceding these examples necessarily require the rule in their 
answers, and they are not placed in connection with the problem in the examples, yet 
the student can readily locate the example or the question to the example. 

What is the interest of $20 for 3 days at 6 per cent. ? 
$20 ■ 
_S 
6)60 
.01.0 the ans. is 1 cent. 

What is the interest of $30 for 3 days at 6 per cent. ? 

$30 $60 for 4 days at 7 per cent. 

3 _^ 

— 240 

6)90 7 

.7l.5=l cent 5 mills. 36)1680(4 cents 6| mills. 

"240 
216 



l2^24/2 



PRACTICAL ARITHMETIC. 71 

What is the interest of $80 for 5 days @ 8 per cent. ? 
$80 * 
5 

8 



36)3200(8.8f the interest is 8 cts. 8f mills. 
288 

320 

288 



4\32/'8 • 

To find the principal, the amount, time, and rate per cent, given. 

Rule. — Find the amount of $1 at the given rate and time, {that is, 
the interest of $l/or the given time added to the $1,) and divide the 
given amount hy it. 

A note which had been on interest 8 years at 5 per cent, amounted 
to $840. What was the principal ? 

8 years. The same @ 6 per cent. 

5 per cent. 8 

lO for $1 for 8 years. ^ 

$1.0^ added. 1.48)840.00(567.56if 

$1740 amt. of $1 for 8 years at 5 per cent. 740 

1000 

Annex two ciphers when the amount is dollars 888 

only. ~rm 

140)840.00($600 principal. 1036 

. S^^ -840 

00 740_ 

1000 

No"w, if the interest is given instead of the amount, di- 833 

vide by the interest, annexing two ciphers when the amt. — 

is doHars only to reduce it to cents. f)ili(lf 



PRACTICAL ARITHMETIC. 



What principal at 7 per cent, will give 
The interest of $1 at 7 per cent, 
is 7 cents. 

9 years. 

63 for 9 years. 
The same at 6 per cent. 



54)80000(1481.48^4^ 
54 

260 
216 



440 
432 

~80 
54^ 

260 
216 



440 
432 

2;54\2T 



The same at 3 per cent. 



27 



in 9 years ? 
63)80000(1269. 
63 
170 
126 

"440 • 
378 
"620 
567 
530 
504 



84A 



260 

252 



27)80000(2962 
54 

260 
243 



.96, 



170 

1 62 

80 
5£ 

260 
243 



170 
162 

8 
2Y 



The prine 



ipal, amount, and time given, to find the rate per cent 



Rule. — If the interest be dollars, add two ciphers to it and divide 
hj the product of the time and principal. 



At what rate 
7 years ? 

Interest, 
600 principal. 
7 time. 

42'00)180.00(4f 
168 00 



0\120 0/2 
0/4200VT 



per cent, must $600 be on interest to gain $180 in 
$180.00. Principal, $600. Time, 7 years. 

Now, a different phraseology would appear to the 
youthful learner entirely to change the construction; yet 
in sense it is precisely the same. Suppose say in this 
way, At what rate per cent, will $G00 amount to $780 in 
7 years ? Now, in this example, by taking the principal, 
$600, from the amount, $78U, we have the interest, $180, 
as in the first example. 



PRACTICAL A R I T II M E T I C. 73 

Tims, Amount, §780 Principal, 600 

Principal, 600 7 years time. 

^ 4200)18000(41 per cent. 

Interest, §180.00 16800 

60 0y42'0 0V7 

Having the principal, interest , and rate per cent, given, to find the 

time. 

Rule. — The same as to find per cent., multiplying the principal 
/ tlie rate per cent, instead of hy the time. 

In how many years will §350 gain §350 at 7 per cent. ? 

Principal, 350 2450)35000(14i| years. 

Rate per cent., 7 2450 



§2450 interest. 10500 

9800 
Prove the last six examples by Simple 

Tntprp«5f- 60\ 700 fl4 



COMPOU,ND INTEREST. 

Compound Interest with most learners seems to present difficulty, 
when, in fact, there is little or no difficulty more than in Simple In- 
terest. Now, you are merely to recollect that Interest is so much 
paid for the use of money, and this interest is reckoned by the year. 
And, at simple interest, say §400 for 1 year at 7 per cent, would amount 
to §28, the §28 being the interest. Now, this §28 interest is not due 
until the borrower has had the use of the §400 1 year ; but then it is 
due, and, if the §28 is not then paid, strict justice requires interest to be 
paid on the §28 as well as on the principal for the next year, and by 
computing interest on this §28 for the next year and all subsequent 
time as well as the principal and all other amounts arising from the 
computation of the increase of interest is compound interest : hence 
compound interest is the charging interest on interest from year to 
year as interest becomes due, as above explained. 

Rule. — Multiply the principal at the rate per cent, as in Simple 
Interest, and add the interest so found to the principal, and it ivill 
give a neiv principal the second year, and so on, until the number of 
inultiplications is equallo'th^e niiniher of years ; observing to cut off 
as many figures from the right hand as there are decimals in both 
factors. -' '-■' 



PRACTICAL ARITHMETIC. 



Note. — When the compound interest only is required, subtract the first principal 
from the last amount. If compound interest is to be computed for many years, the 
operation is long and tedious : and to abbreviate this we append a table, showing the 
amount at 5, 6, and 7 per cent, for 20 years. And the learner can take the number 
"which is under the given rate per cent, opposite the given number of years and multi- 
ply this amount by the pi'incipal, and the product will be the answer. (Cutoff decimals 
as the rule directs.) 



Required the amount of 

500 principal. 

5 



for 3 years at 5, 6, and 7 per cent. ? 

The same at 6 per cent. 
500 
6 



$25.00 interest for 1 year. 
500 

525 principal for 2d year. 
5 



30.00 interest 1st year. 
500 



530 principal 2d year. 
6 



26.25 interest for 2d year. 
525.00 



31.80 interest 2d year. 
530.00 



551.25 principal for 3d year. 

5 



561.80 principal 3d year. 
6 



27.56.25 interest for 3d year. 
651.25 



33.70.80 interest 3d year. 
561.80 



578.81.25 amount. 
500.00 deduct principal. 



595.50.80 amount. 
500.00 



$78.81.25 compound interest. 



500 

7 



.50.80 compound interest. 

The same at 7 per cent. 

572.45 principal 3d year. 

7 



35.00 interest 1st year. 
500 



40.07.15 interest 3d year. 
572.45 



535 principal 2d year. 
7 * 



612.52.15 amount. 
500 



37.45 interest 2d year. 
535.00 



$112.52.15 compound interest. 



572.45 principal 3d year. 



PRACTICAL ARITHMETIC. 



75 



1 
2 
3 
4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 



5 per cent. 

1.05.000 
1.10.250 
1.15.762 
1.21.550 
1.27.628 
1.34.009 
1.40.710 
1.47.745 
1.55.132 
1.62.889 
1.71.034 
1.79.585 
1.88.565 
1.97.993 
2.07.893 
2.18.287 
2.29.201 
2.40.660 
2.52.695 
2.65.329 



6 percent. | 

1.06,000 ' 

1.12.3601 

1.19.101 

1.26.249 

1.33.122 

1.41.851 

1.50.363 

1.59.381 

1.68.947 

1.79.084 

1.89.829 

2.01.219 

2.13.292 

2.26.090 

2.39.655 

2.54.035 

2.69.277 

2.85.433 

3.02.559 

3.20.713 



7 per cent. 

1.07.000 
1.14.490 
1.22.504 
1.31.079 
1.40.255 
1.50.073 
1.60.578 
1.71.818 
1.83.846 
1.96.715 
2.10.485 
2.25.219 
2.40.984 
2.57.853 
2.75.903 
2.95.216 
3.15.881 
3.37.993 
3.61.659 
3.86.976 



The first number may be omitted to ob- 
tain the interest until that number is more 
than 1. 



Now, if you desire the amount at 
compound interest, multiply the prin- 
cipal by the whole number as found 
under the rate per cent, and opposite 
the time ; but if the compound interest 
is only desired, omit the first figure, 
for instance, the interest for 3 years 
at 5 per cent. You find opposite 3 
and under 5 per cent. 1.15.762. 

1.15.762 interest from the table. 
500 amount. 



578.81.000 



$78.81 is the interest and $578.81 the 
amount. 

At 6 per cent. 
1.19.101 

500 



$595.50.500 amount. 
500 



$95.50.500 interest. 

At 7 per cent. 
1.22.504 
500 



$612.52.000 amount. 
500.00.0 



$112.52 interest. 



DISCOUNT, COMMISSION, INSURANCE, AND 
BROKERAGE. 

Discount is an allowance made for the payment of ready cash, 
and Insurance, Commission, and Brokerage are amounts paid to 
insurers, factors, or commission men, and brokers, at such rate per 
cent, as is agreed on, &c. j. 

Note. — The most of our arithmetics blend discount and present worth, although 
there is a striking difference. Discount may be reckoned without regard to, or at least 
without any definite, time ; but present worth is the amount after discount has been 
allowed agreeable to the per cent, and time ; and each of the above has been fully 
explained in the Single Rule of Three and Simple Interest. In fact, they are synony- 
mous to those rules, although we again present an example in each. 

What is the discount of $670 at 7 per cent. ? 

$670 
7 



$46.90 



76 



PRACTICAL ARITHMETIC. 



YvHiat is the insurance on a house worth $2260 at 1| per cent. ? 



2260 

2260 at 1 per cent. 

1130 at J per cent. 

565 at I per cent. 



• $39.55 at If per cent. 

What is the commission of the sale of $3640 of goods at 2 J pei 

cent. ? 

$3640 
21 



T280 at 2 per cent. 
910 at J per cent. 



$81.90 at 21 per cent. 

What is the brokerage of $4326 at IJ per cent. ? 

$4326 



43.26 at 1 per cent. 
10.81.5 at i per cent. 



$54.07.5 at IJ per cent. 

Now, present worth is found as in finding the principal having 
time, rate per cent., and amount given. 

Rule. — Find the aiit)unt of $1 for the time and given rate, and 
divide the given sum hy it. 

Note. — If the sum or araount is dollars only, annex two ciphers before dividing. By 
subtracting the present worth from the whole sum or amount, of course will give the 
discount. Amount is $1 and interest added, as the amount of $1 for 1 year at 7 per 
cent, is $1.07, for 2 years $1.14, for 8 years $1.21, and so on. 



PRACTICAL ARITHMETIC. 



What is the present worth of $2280 due 3 years hence, at 7 per 
cent. ? 

$1 121)2280.00(1884.29^ present worth. 



T 


121 


7 1 year. 


1070 
968 


3 
, 1.21 3 years. 


1020 
968 

520 


$2280.00 amount. 
$1884.29^V present worth. 


484 

360 

242 


§395.70i3^ discount. 


1180 

1089 



91 

What is the present worth of §552.50 due 5 years hence, at 6 per 
cent. ? 

5 years. 130)552.50(425 present worth. 

6 520 



325 $552.50 amount. 

$1.30 amt. of $1 for 5 years. 260 ' $425.00 

^650 $127.50 discount. 
650 

What is the present worth of $420 due in 3 years and 9 months, at 
6 per cent. ? 

1 1225)420000(342.8511 

6 3675 



6 5250 

3 4900 



18 for 3 years. 94^0 

3 for 6 months. 



1.5 for 3 months. 1^500 

9800 

22.5 



I 100 ' ^^^^ 

I ^'■' 6125 

1.22.5 25\ 875/35 

25A225UI/ 

In this example we have to add three ciphers, as the dividing term has mills. The 
reason — if the sum is dollars only two ciphers must be added — is, because, as in Pro- 

j portion, the terms must be of the same denomination, therefore the sum $420 has to 
be brought to mills to be in proportion with the dividing term, which is $1.22.5. It i.« 

I customary for banks to find the interest on the time with three additional days, and 

1' 6 



78 PRACTICAL ARITHMETIC. 

term this interest discount, which is deducted from the note. Now, by law a person 
is not obliged to pay a note until three days after it becomes due, which are called 
days of grace. You can find the interest of $100 instead of for $1, and add the $100 
to the interest found for the given time, and divide the principal by it, which will 
produce the same result as the $1 and the interest. 



LOSS AND GAIN. 

Loss AND Gain is a very important rule to trading men and mer- 
chants to ascertain their gain or loss on any thing bought or sold ; 
and every person should be competent to ascertain the amount of 
gain or loss in trade, also the gain or loss per cent. The amount of 
loss and gain in general is by subtracting the less from the greater, 
which will give the gain or loss as the case may be, &c. 

' The Gain or Loss in General. 

Bought 20 bbls. flour for $6.50 and sold the same for $8 per barrel. 
What is the gain per barrel and gain on the whole ? 

Sold for $8.00 
Bought at 6.50 



$1.50 gain per barrel. 
20 bbls. 



.00 gain on the whole. 

Bought 600 bushels wheat for $850. I wish to gain $120. How 
much must it be sold per bushel ? 

Cost, $850 600)970.00(1.611 per bus. • 

I wish to gain 120 600 Proof. 

must sell for. ^700 1.61f 

3600 600 

"1000 96600 at 1.6L 

600 400 at f. 

i^ljii^d $970:00" 

To find what is gained or lost per cent. 

Rule. — See tvhat is the gain or loss hy subtraction : annex two 
ciphers to gain or loss, that is, if the dividend is dollars only ; hut 
if the dividend is dollars and cents, and dollars only in the divisor, 
that is the gain or loss : the ciphers need not be added. Then divide it 
hy the sum which is laid out. 



PRACTICAL ARITHMETIC. 79 

Operation — Grain . 

Bought 40 yards prints at 35 and sold the same at 42 per yard. 
What is the gain per cent. ? 

Sold at 42 35)700(20 per cent. 

Cost, 35 70 

Gain per yard, 7 cents. 

(7 is 1 of 35 = f is 20 per cent.) 

Eonght 4 hhds. sugar weighing 3500 fbs for $315, and sold it at 
7^ cents per ft). What was the gain or loss per cent. ? 



3500 


Gave $315 
Sold, 262.50 


315)5250(161 loss per cent. 
315 


24500 
1750 


Loss, S52.50 


2100 

1890 


§262.50 sold. 


mYMi 



To find how any tiling must he sold to gain or lose at a given rate 

'per cent. 

Rule for Gain. — Add the given rate per cent, to $1 and multiply 
the cost price hy it. Out off ciphers, as directed in Interest. 

Bought sugar, cheese, prints, ribbons, &c. at 8J cents. For what 
must they be sold to gain 20 per cent. ? 

Operation, 
Now, 8J cents is 8 cents 5 mills. 
Hence, 8.5 by 1.20. or if $8.50 per hundred. 

1.20 $1 to 20 is 120 



1700 17000 

85 850 



10.2,00 = 10 cents 2 mills. $10.20,00 $10 and 20 cents. 

Now, 10 cents 2 mills is 10|, so $10.20 $10|, both the same. 

Rule for Loss. — Noiv^ for loss subtract the loss per cent, from $1 
and multiply the remainder by the purchase price. 



80 



PRACTICAL ARITHMETIC. 



Bought goods at 95 cents, and, finding them damaged, am willing to 



lose 12 per cent. 

$1.00 

12 rate per cent. loss. 

88 remainder. 
95 purchase. 

440 

' 792 



or if $95 



$100 
12 

88 
95 

440 

792 

$83^60 



83.6.0 = 83 cts. and 6 mills, selling price. 

Bought sugar, cheese, coffee, yds., &c. at 12 cents, but, wishing to 
close business, am willing to lump the stock at 5 per cent, loss, 

1.00 or if 12 dollars 100 

5 5 



95 
12 

11.4.0 = 11 cents 4 mills or 11|. 



95 
12 

$11.40 cents. 



Bought muslin at 32 cents per yard, and sold it at 37J- cents per 
yard. What is the gain per cent. ? 

Sold for 37J 32|0)550|0(17^ gaJn. 

Bought for 32 320 

— 230 

5J which is 5 cents 5 mills. 224 

2\ 6 / 3 

The same bought at 37J cents and sold at 32. 

Gave 371 375)5.500(14f loss. 

Sold, 32 375 



5J is 5 cents 5 mills. 
37i is 37 cents and 5 mills. 



1750 
1500 



LM/3T5V¥ 



FELLOWSHIP. 

Fellowship is used by persons trading in company with each 
other as partners, to ascertain their respective shares of the gain or 
loss. It is of two kinds, single and compound. 

Single Fellowship is used when the capital is employed in trade an 
equal term of time, but the shares invested by the partners are of 
various amounts. 



PRACTICAL A III T II M E T I C. 81 

Rule. — Multiply the gain or loss by each person s share of the 
stock, and divide each product by the ivhole stock or capital invested. 

A firm of three persons, Davis, Mason & Brewton, gained $1200 : 
Davis put in $1500, Mason $1200, Brewton $900. What is each 
man's share of the gain ? 

Davis, $1500 Gain, 1200 
Mason, $1200 1500 Davis's stock. 

Brewton, $900 600000 

— ^ 1200 

All, $3600 3600)1800000(500 Davis's gain. 
Gain, 1200 18000 



1200 Mason's stock. qq 



3600)1440000(400 Mason's gain. 
14400 



Proof. 



0^ Davis, $500 

Gain, 1200 Mason, 400 

900 Brewton's stock. Brewton, 300 

3600)1080000(300 Brewton's gain. ^TZTZ 

10800 ' ^^^^^ 

00 



Note, — This is a beautiful rule, and school-commissioners apportion the different 
teachers by this rule ; also school-teachers may apportion among their employers ac- 
cording to the days each patron has sent. Taxes are assessed, also a bankrupt's estate is 
divided among his creditors, &c. 

Although the rule laid down and the example presented is not fully stated and 
solved according to Proportion or Single Rule of Three, it is nevertheless proportional 
in effect. In fact, you have no example or rule laid down as yet that is not an ab- 
breviation of Proportion or Single Rule. 



COMPOUND FELLOWSHIP 

Is when the several shares of stock and the term of time are both 
unequal. 

Rule. — First, multiply each share of the stock by the time, and 
add the products together ; then multiply each mans product by the 
gafn or loss, and divide each of these products by the sum of the pro- 
ducts. 



82 PRACTICAL AEITHMETIC. 

Smith & Hill trade in company: Smith put in $2000 for 15 
months, Hill $1600 for 11 months. Thej gained $900. What is the 
gain of each ? 
Smith, $2000 

for 15 months. Hill, $1600 

1^^^^ for 11 months. 

^^^^ * 17600 Hill's product. 

30000 Smith's product. 
17600 Hill's product. 

i7600sumofbothproducts.™'s P^^^^^^^ l^^^J^^ ^^.^^ 

Smith's product, 30000 — 

900 gain. 15840000 Hill. 



27000000 Smith. 

47600)27000000(567.22.6iff Smith's gain. 
238000 
320000 Hill's gain. 

285600 47600)15840000(332.77.33^ 

■144000 142800 

333200 156000 

-108000 142800 

95200 ' 132000 

"128000 95200 

95200 368000 

333200 



328000 

285600 348000 

400\42400/'l0 6 Oo3200 
400;47600VTl¥ 



D ^ ^ .7. . 148000 

Proof of the gain. IAQqao 

Smith's, $567.22.61^1 ^^^^^^ 

Hill's, $332.77.33^«^ m)ifiU 



$900.00.0 



Now, if a firm make a stock a certain time and each partner may put in or draw 
out, you first multiply the stock put in for the time up to the time when he puts in 
again or draws, as the case may be, subtracting or adding for the next time, and so on. 

For a fuller explanation, we present an example showing the operation for a limited 
time, in which the partners put in and draw out at different times. We take the 
following from Smiley's Arithmetic, for the reason that we have rarely, if ever, met 
the pupil who solved the question without assistance, although it is by no means a 
diJBficult one. 

A, B, and C made a stock for 12 months : A at first put in $580, 
3 months after $100 more. B at first put in $1000, after 9 months 



PRACTICAL ARITHMETIC. 83 

$200 more. C at first put in $480, 3 months after he took out $300, 
2 months after put in $500, 3 months after he took out $400, 1 
month after put in $1000. They gained $2108.44. What is each 
one's gain ? 

A at first $580 for 3 mos. 1740 B at first $1000 for 9 mos. 9000 
Then 100 Then 200 

680 for 9 mos. 6120 1200 for 3 mos. 3600 



7860 A's product. B's product, 12600 

C at first $480 for 3 mos. 1440 
Then took out 300 

180 for 2 mos. 360 ^ow, A's product, 7860 

Then put in 500 B's " 12600 



680 for 3 mos. 2040 

Took out 400 



C's " 7960 



All, $28420 



280 for Imo. 280 • 

Put in 1000 Now you have the 

question ready for 



1280 for 3 mos. 3840 statement in Single 

Rule of Three, or by 

12 mos. 7960 C's product. ^^^ preceding rule, 

to multiply each 
man's product by the gain and divide the product by the sum of the products, which 
is nothing short of the Single Rule of Three : thus, 
Sumof products. A's product. Gain. 

$28420 : $7860 :: $2108.44, which will give A's gain. 
Sum. C's product. Gain. 
$28420 : $12600 : : $2108.44, which will give B's gain. 

Sum. C's product. Gain. 
$28420 : $7960 : : $2108.44, which will give C's gain. 



FRACTIONS-VULGAR AND DECIMAL. 

Under this head we intend, so far as we are permitted in this 
abbreviated and systematized arrangement of practical rules and ex- 
amples, to be full, or, at least, more so than on any former rules or 
examples ; for which we presume we are warranted in so doing from 
several considerations. First, from the fact the Decimal or Frac- 
tion will express that which we are unable elliptically or with good 
euphony to express without their use. 

Secondly, because Fractions and Decimals in their various uses 
and applications present the highest and most skillful science of 
mathematics ; and he who is proficient in the various powers and 
effects of Fractions and Decimals can hardly be unproficient in whole 
numbers. They are the strongest arm and power of mathematics. 



84 PRACTICAL ARITHMETIC. 

They are the varnishing and polishing of the whole arrangement of 
mathematics. They act as a purifier and refiner to the whole beauty 
and scheme of mathematical science, &c. 

Thirdly, ours is a great country, — a decimal or fractional country : 
our standard currency is decimal or fractional. Then why not con- 
form to it in the highest sense of the terms, &c. ? The decimal or 
fractional system of the United States is the most convenient system 
in the world; and yet, after having adopted and legalized it, our 
people to some extent in some sections still exhibit an inclination ta 
adhere to the old colonial system of pounds, shillings, and pence ;' 
and, as this latter currency difiers so much in value in the different 
States that he who still adheres to it is forced to keep up a running 
process of currencies in his mind, and, unless he be a good reckoner, is 
liable to be imposed upon after all of his constant perplexity. We speak 
mostly one language, and we should. feel proud of this language, and 
surely we would so feel if we understood it properly and correctly, — 
which can be is to he, and the sooner the better. We should likewise 
have one currency only, equally understood by all, as the law of this 
happy land has contemplated. We know there is an objection, or, at 
least, a remissness by some teachers in the instruction of decimals and 
fractions. We have witnessed this before our board of commissioners 
in the examination of teachers ; also we have had pupils, frequently, 
too, to remark to us 'Hhey did not think fractions much account — done 
no good, and, further, that their former teacher taught so too, and 
therefore jT s^^jope(i them,'' &c. Now, but one idea only ever presented 
itself to my mind as to teachers of this class : they very truly though 
unintentionally spoke the truth. They understood them not, and 
consequently they give them but little importance; and the admission 
of the teacher of his allowing the pupil to skip, as it is termed, frac- 
tions on account of inutility, is a palpable admission of negligence or 
ignorance. 

This subject is worthy of the attention of every teacher throughout 
the land, not only in relation to a certain branch in mathematics or 
arithmetic, but it is. equally true concerning every branch taught in 
our schools. And, further, it is the imperative duty of each and 
every teacher to strictly confine and impress with his utmost efforts 
the strongest and most practical parts included in any branch he at- 
tempts to teach ; and he who is so stupid as not to perceive the stronger 
and most available parts in instruction is unworthy of the support 
or respect due a competent man. 

All departments or branches in education, from the lowest to the 
highest, have each their stronger and more practical parts ; and he 
who teaches to this end, or who presents a text-book in the greatest 
degree to this end, is he that is disseminating or scattering, as it were, 
bread upon the waters, to be taken up many days hence. Fractions 



PRACTICAL ARITHMETIC. 85 

and Decimals, m connection with the Single Rule of Three or Pro- 
portion, are the mathematician's compass, the practical business-man's 
every thing. They are his steam, sails, oars, &c. ; and he who 
thoroughly understands them is a competent business-man, &c. 

Note. — "We wish to remind the learner that the Analytical Dictionary of this work 
is as applicable and as beneficial to you in your mathematical operations as in any other 
department. Have you digested the complex or technical terms in those questions or 
notes to fully comprehend them ? If you have not, tou have been remiss in your 
application. To remind you and the teacher likewise, that every complex word in 
arithmetic should be analyticaMy considered, &c. : hence, will you analyze the follow- 
ing terms: vulgar fractions, proper, improper, compound, mixed, numerator, de- 
nominator, &c. ? 

Teacher. What is a vulgar fraction .? 

Student. A vulgar fraction is a part of a whole number. 
T. How expressed ? 

*S'. By first calling the upper number and then the lower : thus, J 
one-fourth, J- one-eighth, ^ five-sixteenths. 

T. Wliat is the upper part of a fraction called ? 

aS^. The upper part is called the numerator, 

T. What does it show ? 

aS'. It shows the part of a whole number expressed by the fraction. 

T. What -is the k)wer part of a fraction called ? 

S. The lower part of a fraction is called the denominator. 
T. What does it show ? 

S. It shows the number of such parts contained in a whole number. 

T. How many different situations may fractions be considered ? 

S. Fractions may be considered of four kinds. 

T. Will you name them ? 

S. Proper, improper, compound, and a mixed fraction. 

T. What is a proper fraction ? 

S. The numerator of a proper fraction is less than the denomi- 
nator ; as, 4, g-, yI) 2T> oT' *'^^' 

T. What is an improper fraction ? 

S. An improper fraction has its numerator greater than the de- 
nominator ; as, f, f. If, H, &c. 

T. What is a compound fraction ? 

S. A compound fraction is a fraction of a fraction ; as, f of J of 
|;fofJ, &c. 

T. What is a mixed fraction ? 

aS'. a mixed fraction has a fraction attached either to its numerator 
or denominator. 

T. What is a mixed number ? 

S. A mixed number is when a fraction is annexed to a whole 
number. 

T. What does every vulgar fraction denote ? 

S. Every vulgar fraction denotes division. 

Note. — The numerator is the dividend and the denominator the divisor : therefore 
in all proper fractions the numerator or dividend is less than th^ denominator or di- 



86 PRACTICAL ARITHMETIC. 

visor, conseq-aeiitly the sum is less than 1 ; but an improper fraction has the numerator 
the largest, and the sum must be greater than 1. If the numerator and denominator 
are the same, the sum is 1. 

T. What is the rule to reduce fractions to their lowest terms ? 

S. First divide the denominator bj tlie numerator ; then take the 
numerator for a dividend and the remainder for a divisor, and so on, 
alternately, until no remainder is left ; then divide the numerator and 
denominator by this number. 

Operation, 

No. 1. No. 2. 

Reduce ||f to its lowest terms. Reduce 3^^ to its lowest terms. 

288)690(2 984)3648(3 

576 2952 



114)288(2 696)984(1 
228 696 

60)114(1 288)696(2 

60 576 



54)60(1 120)288(2 

54 240 

6)54(9 "48)120(2 

54 96 

You see that 6 leaves no remainder, Equal divisor, 24)48(2 




48 



therefore must be a common divisor. 

Thncj 6^288/48 times in 228 24\ 984 Z' 41 

J.UUi3, 6/690VH? times in 690 2i/3648\152 

Now, this process is the method of reducing all remainders to their lowest terms. 
The remainder after you have run the dividend out is the numerator, and the divisor or 
sum you have been dividing by is the denominator. As in all the former examples, the 
remainder has been reduced to its lowest terms in the above way. 

To find the fractional parts of numbers. 

Teacher. How do you find the fractional parts of numbers ? 

Student. Divide the given number by the denominator of the frac- 
tion, and if there be a remainder place the divisor under it. Also 
reduce to its lowest terms, if necessary. 

What is J of 144, and J of the same ? also J ? 
J)144 i)144 J)144 

48 = I of 144. 86 = i of 144 18 = J of 144. 

What is ^ of 1164 ? also ^V ? 



PRACTICAL ARITHMETIC. 87 



13)1164(89^ = 
104 


= ^ of 1164. 


60)1164(19-1 = - 
60 


124 
117 




564 
540 


A 




24)60(2 
48 




Now, mil 


12)24(2 
24 



- When the numerator is more than 1. 

Teacher. When the numerator is more than 1, how do you proceed ? 

Student. Multiply the given number by the numerator, and divide 
the product by the denominator. 

1. What is I of 842 ? 2. What is H of 3840 ? 

842 3840 

4 numerator. 11 numerator. 



Denominator, 19)42240(2223^ 



Denominator, 5)3368 * 3g 

673f = I of 842. f^ 



44 

38 
60 

57 



To reduce improper fractions to tvhole or mixed numbers. 

Teacher. What is the rule to reduce an improper fraction to a whole or mixed 
number ? 

Student. Divide the numerator by the denominator, and if there 
be a remainder reduce to the lowest terms, and place it to the right 
hand of the quotient number. 

1. Reduce ^ff* to a whole or 2. Reduce ^ff^ to a whole or 

mixed number. mixed number. 

19)1384(72i| Answer. 25)4628(185/^ Answer. 

133 25_ 

~54 212 

38 200_ 

H "128 

125 



8q practical arithmetic. 

To reduce mixed numbers to improper fractions. 

Teacher. What is the rule to reduce mixed numbers to improper fractions ? 

Student. Multiply the whole number by the denominator of the 
fraction, and add in the numerator. 

1. Reduce 780|| to an improper 2. Reduce 325j^. 
fraction. 

780 325 

32 11 



1560 
2340 



3581 add in the 6. 



26 add the numerator. H 

24986 ^^^^ ^^^ *^® preceding rule serve to act, 

— as it were, by reversion, one being the re- 

32 verse of the other. 

To reduce compound fractions to a single fraction. 

Teacher. What is the rule to reduce a compound fraction to a simple one ? 

Student. Multiply all the numerators together for a new numerator, 
and all the denominators for a new denominator : if necessary, reduce 
to their lowest terms. 

1. Reduce | of J to a simple fraction. 

fx I = ^V Answer. 

2. Reduce J of f of ^ of ^ to a 3. Reduce f of f of 8f . 
simple fraction. Now, 8| is f 

X ,V == ^fix Ans. ^ ' 

xins. 



i X 1 X J-, X ,V = efk An 
16 


s. 

Hence | 


'7 ^ 

XI 


"6 ' 

X 


"^ 3 


12 
192 


7 
3 






189)780(4^ 
756 


4 

768 


21 

9 






DilUA 


8 


189 









6144 

Note. — To find the greatest common divisor of two or more whole numbers, as well 
as a common divisor for a proper fraction, divide the greater number by the less, and 
if there be any remainder divide the divisor by this remainder, and so on, continuing 
to divide the last divisor by the last remainder until there is no remainder. 

What is the greatest common divisor of 39 and 91 ? 
Now, 39)91(2 or, if it were a fraction, 

78 39)91(2 If = I 

— 78 

Ans. 13)39(3 Jg^g^^g ^^,,(3 

"^^ 39 



PRACTICAL ARITHMETIC. SO 

To find the least common multiple of a number. 

XoTE. — The above properly comes under the head of fractions. A number that cnn 
be divided by two or more numbers is called a multiple of those numbers, and numbefs 
are either prime or composite. A prime number is one that cannot be produced by 
multiplying any two or more numbers together; as, 2, 3, 5, 7, 11, 13 ; but a composite 
number is the product of two or more "numbers ; as, 8, 10. 8 is the product of 2 and 
4, 10 of 5 and 2. 

What is the common multiple of 6, 8, 10, 18 ? 
2)6 8 10 18 

The operation of this is very simple, — simply to 
divide by a number that will divide two or more of 
the numbers, and so on, divide their product by a 
number that divides two or more. In the first, 2 
divides all the numbers, but in the second, 3 only 
divides two of them. The numbers 4 and 5 are 
brought down, and the products and divisors are 
multiplied, which give their least common multiple. 
Also, this rule would serve to present some ingenuity 
and skill, as questions of such a nature are properly 
solved by the same process. 



Suppose you were ordered to get a pole, and wished to know how 
long to cut it, and it is to be the shortest that can be exactly measured 
by a pole either 6 or 10 feet long. What must be the length of the 
pole you get ? (Olney's Arithmetic.) 

Operation. 
2)6 10 Or measured by a pole either 8 or 

r^ 14. 



3)3 


15 9 


1 


4 5 3 
3 




15 
4 




60 
3 2d divisor. 




180 

2 1st divisor. 




360 Answer. 



5 2)8^ 

4 7 

28 



2 



15 

i - 

30 feet. 

1 ■ Now, the least common multiple is also 

I the least common denominator, which is 5g feet Ans. 

I obtained by the same process. ' 

1 To find a common denominator. 

Teacher. What is the rule to find a common denominator ? 

u Student. Divide the denominators by any number that will divide 

1 two or more without remainder, and continue the division of their 

' quotient or quotients till no two numbers can be divided ; multiply 

the quotients and the divisor or divisors for the least common denomi- 



1. 



90 PEACTICAL ARITHMETIC. 

nator, and divide this by each denominator, and multiply each quotient 
by its own numerator, and the new several products will be the new 
numerators, which place over the common denominator. 

1. ' Reduce f, ^, f to a common denominator. » 



Now, 2)8 12 6 are the denominators. 


Then, 8)48 = 6 x 6 = |f 


3)4 6 3 
4 2 1 








And 12)48 = 4 X 4 = if 


2 

8 
3 

24 




And 6)48 = 8 X 2 = If 






2 






48 common 


denominator. 




2. Reduce |, ^^o, 


3^, and ^ to a common denominator. 


Now, 5)^ 20 10 


15 


5)60(12 X 4 = If 


2)1 4 2 


3 


60 4 


12 1 


3 


48 


3 




20)60( 3 X 9 = g 


6 




60 9 


2 




27 


12 




10)60( 6 X 7 = M 


5 




60 7 


60 common denominator. 


42 

15)60( 4 X 4 = i| 
60 4 


(Smiley's Arithmetic.) 


It is necessary to divide by prime num- 




bers to find the least 


common multiple or 


16 



41 



the least common denominator ; otherwise the answer will be too large. 

To reduce fractions of a less denomination to that of a higher de- 
nomination^ hut retaining the same value. 

Rule. — Multiply the denominator of the given fraction with all 
the denominations between the given fraction to which it is to he re- 
duced, which reduce to a simple fraction and then to its lowest terms. 

1. Reduce J of a cent to the fraction of a dollar. 

I X T^ = ^ of a dollar. 

2. Reduce f of a penny to the fraction of a pound. 



PRACTICAL ARITHMETIC. 91 

i of il of .V = 1)^0 = £^' 

12 

^^ Now, the denominations between a penny 

9TQ and a £ are 12 and 20 ; as, 12 pence Is., 20s. 

% 1£. 

960 

3. Reduce | of a pint to the fraction of a hogshead. 
I of -J of J of J3 = |)^4_^ =1^^. 

63 
4 

Between a pint and hogshead are 2, 4, 

252 and 63 ; as, 2 pints 1 qt., 4 qts, 1 gal., 63 

O gals. 1 hhd. And you only have to observe 

• the' same principle in any other denomina- 

504 tion. 

8 



4032 



To reduce the fraction of a higher denomination to a less denomina- 
tion, but retaining the same value. 

Rule. — Blultiply tJie numerator hy all the denominations between 
the given fraction and the denomination to which it is to be reduced, 
and place the product for a new numerator over the given denomi- 
nator, ivhich reduce to its lowest terms. 

Note. — This and the preceding terms are in reversion of each other. In the i5rst, the 
denominator is multiplied by the denominations betwenthe given denomination and that 
to which it is to be redviced, while in the latter the denominations are multiplied by 
the numerator, then reduced to its lowest terms. 

What part of a pint is gf^ of a bushel ? 

^ X f X f X 1 = iM. 192)320(1 

192 

I28yi92(l 
M)iM(l Answer. 128 



64)128(2 
128 



What part of a penny is j^ of a pound ? 

1^ X ¥ X \« = ^^)\m = If Answer. 



To reduce a fraction to its proper value. 

Rule. — 3Iultiply the numerator by the next lowest denomination, 
and divide by the denominator. 



92 PRACTICAL ARITHMETIC. 

1. Reduce ^| of a dollar to its proper value. 

13 

100 

16)1300(811 cents. 
128 

~20 
16 

Ma 

2. Reduce ^^ of a dollar to its proper value. 

16)900(56J cents. 
80 

100 



4A6U 

3. Reduce | of a dollar to its proper value. 

8)700(871 cents. 
64^ 

60 
56 

m 

4. Reduce f of a dollar to its proper value. 

3 X If = 300. 4)300 

75 cents. 

5. Reduce f^ of an acre to its proper value. 

^ X f = f§ 16)160(10 perches. 

16)20(1 R. 10 perches. 16 

16 -TT 



Now, it is necessary for the user of fractions to have a double view of the terms 
expressed, not only so in fractions, but in all education. When we say 13-16ths, the 
mind should say, within itself, that is 81^ cents; 9-16ths, that is 66] ; 7-8ths, that is 
87i cents ; or, if we say 81], that is 13-16ths ; 56], that is 9-16ths ; 87J, that is 7-8ths, 
making a double use,— one expressed, and the other understood or expressed by reflec- 
tion in the mind. This double use of numbers or any other branch to education is 
the very life and progress of education. 



PRACTICAL ARITHMETIC. 



93 



To reduce any given value or quantity to the fraction of any greater 

denomination. 

Rule. — Reduce the given sum to the loivest denomination men- 
tioned for a numerator, then multiply all the denominations of ivhich 
you tvish to maJce it a fraction to the same denomination as the nume- 
rator for a denominator 



1. Reduce 81J cents to the fraction of a dollar. 

Which is ^■*. No number will divide 81 J: therefore bring it to 
fourths ; also the 100, the denominator ; as, 81 J is ^*. 



81i 
4 



100 
4 



Then, |f)fM(ii 



Ans. 



325 



400 



2. Reduce 561 cents to the fraction of a dollar. 
That is ^, which is m S^S: 



225)400(1 
225 



175)225(1 
175 



50)175(3 
150 



25)50(2 
50 



[f)fM(^ Ans. 



3. Reduce 9s. 4^?. to the fraction of a pound. 



Now, 9s. 4(i. c£l None can be so negligent or unthought- 

^2 20 ^^ ^^ ^^® ^^^^ ^^^^ ^^^ rules in arithmetic 

^ - — are frequently just a reversion, as the fwo 

is 112(f. is 20.?. preceding rules ; and none who are pro- 

Then JJ^ is <£^ 12 ficient in arithmetic are ignorant of the 

-*'^ ^^' _ fact that all the diflacultv with the learner 

112)240(2 is 240<i. is in the bearing or construction of the 

224 question : hence the importance of con- 

- — — 1 o rr \ - struction and application as urged previous 

16)112(7 if/liottA-nS. to this. Learner, you are not to cramp the 

W^t mind ; you must allow it to expand, assist 

and cultivate it so to act and give your con- 

ceptive powers full scope and range. Avail yourself of all the benefits derived by 

comparison of questions. See the change of construction. Strengthen the mind by 

the comparison of rules in reversion of construction and operation, and in each step 

you advance you make the next more rapid and easy. 



94 PRACTICAL ARITHMETIC. 



ADDITION OF VULGAR FRACTIONS. 

Teacher. Will you repeat 'the rule for the Addition of Fractions ? 

Student. Reduce the fractions to a common denominator, and add 
the numerators together for a numerator to the common denominator. 

T. If mixed numbers are given, how will you proceed ? 

S. Use the fractional part only as the rule above, until the frac- 
tions are added together, and then add the whole number by simple 
addition, observing to add the gain, if any, in the fractions to the 
whole numbers. 

T. When different denominations are given, how will you proceed ? 

S. Find the proper value of each separately, and add them together 
by compound addition. 

T. If all the denominators are alike, how do you proceed ? 

S. Add all the numerators together, carrying one for each time 
the denominator is equalled, and then place the product or remainder 
over one of the denominators. 

1. Add ^, ^, and A together. ^4^ + 3^ + ^ = i| = l Ans. 

2. Add Jg, J-, J, and J together. 

4)16_8_^ r^^ ^lg^-^g _ 1 X 1 = 1 

2)4 2 12 2^ 

Vm ?)^^ = 2x1 = 2 

2 4)16 = 4x1 = 4 

"5" 2)16 = 8x1 = 8 

4 . — 

16 common denominator. * ^^ 

Note. — Double use for l-16th is 6|- 
for l-8th is 12| 

for l-4th is 25 Proof. 

for J is 50 93| is 15-16ths, or 15-16ths is 93f. 

93| 

3. Add 5|- , 4f, and | together. 

4)|_i_tthe fractions. rpj^^^^ ^^^ _ 14 X 3 = 42 

"^ 2 ^ f)56 = 8 X 2 = 16 

14 8)56 = 7x5 = 35 

_1 — 

56 common denominators. Ans. 5^ 



PRACTICAL ARITHMETIC. 



95 



The 5 and 4 are the whole numbers. 

5 
4 

Ifl fractional amount. 

lOfl Ans. 



II is Ifi. 



56)93(111 
56 



The small figures, 1, 2, 3, 4, as above 16, 8, 4, 2, also 1, 2, 3, as above 4, 7, 8, 
indicate the denominators in examples 2 and 3, by which the common denominator is 
divided and their products multiplied by their respective numerators. 

4. Add J of an acre to f of a rood. 

1 X i = |. Or, 1 of f of \o == 160. 



3)4(1 R. 36^ P. 
3^ 

J rood remainder. 
Then, i and f. 
3 
7 

21 common denominator. 

8)21 = 7x1= 7 

7)21 = 3 X 4 = 12 

Then, |f x f = W. 
21)760(362^ poles or perches. 

63^ 

130 

126 



3)160(531 perches. 
15 



10 

9 



40)53J(1 R. 13 J per. 
40 

181 
Then, f of f = ^\ 

7)160(22f per. 
14 

^20" 
14 



R. per. 
13^ 

22< 



1-131 



Ans. IR. 862*1 P^^- 



This example is applicable to add fractional parts of any of the weights, measures, 
&c. ; also it applies to subtraction of the same, by subtracting the less from the 
greater. 



To add fractional parts of weights, measures, ^c. 
Add ^ of a day, f of an hour, ^ of a minute together. 



9a 



PRACTICAL ARITHMETIC. 



^ 0^ a day, or of 24 hours. 
7)24(3 hours. h. min. sec. 



21 

3 hours. 
60 



25 
13 



42| 

20 

32^ 



7)180(25 minutes. 3 
14 

"40 
35 

60 

7)300(42f seconds. 
28 



39 



'9T 



20 
14 



Add f and {-^ 
91 



7)91 
13)91 



13 X 6 = 
7x4 = 



78 
28 



91)106(lif 
91 



I of an hour, or of 60, min. 
60 

2 ■ 

9)120(13 minutes. 
9 

"30 

27 

3 minutes. 
60 

9)T8"0(20 seconds. 
18 




5^ of a minute, or of 60 sec. 



60 

7 

13)420(323^ seconds. 
39 

~^ 
26 



To subtract fractional parts of weights, measures, ^c. 

From f of a bushel take f of a quart, 
f of a bushel, or of 4 pecks. f of a quart, or of 2 pints. 

f X f = V pks. f X f = I pts. 

8)12(1 pk. 4 qts. |)|(f pints. 

pk. qts. pts. 



_8 

T 

_8 

8)32(4 quarts. 



14 



Ans. 1 



3-1 



To subtract a fraction from a mixed number or from a whole number. 
From $370 take $51^. From 428 take 126if. 



$370 
51.^ 



428 
126i| 



$318if Answer. 



301j^ Answer. 



PRACTICAL ARITHMETIC. 97 

Note. — In the subtraction of a fraction from a w^ole number, as in the above examples, 
you take the numerator from a borrowed denominator 1, which is equal to the de- 
nominator, as in the examples 4 is taken from 20, and 12 from 17, as 20-20ths or 
17-17ths is 1. 



MULTIPLICATION OF VULGAR FRACTIONS. 

Rule. — Multiply all the numerators of the given fraction tog ether, 
for a neio numerator^ and all the denominators for a new denomi- 
nator, and, if necessary, reduce to its lowest terms. 

Note. — Whole or mixed numbers to improper fractions, and compound fractions to 
simple ones. Now, a whole number, as 9, is reduced to an improper fraction by a 
line and 1 under it ; as, |. 



1. Multiply I by 3^. 

Thus, 

2. Multiply 7f by 8f . 

7f 



I X ^ = f f reduced is ^ Ans. 



8| 
8 



69 



Then, ^^ X f = 



3. Multiply 6^ by 7§ and 15. 

6 7 

14 32 

24 14 

6 21 

9 21 

93 245 

14 31 

Then, If X W X '{ = '^m^ 



\2\( 7 
733VTT* 



56)3588(64^5 
336 

"228 
224 



D^fe 



448)341775(762f| Ans. 
3136 



2817 
2688 

"l295 

896 



DMKM 



Note. — This rule can be abridged by cancelling, although it can only be done when 
the numerators and denominators contain some factors that are equal ; and yet some 
j have attempted to invest this method of cancelling with mathematical skill and im- 
portance which it does not possess. 



By cancelling. 

^ Rule. — Strike out as many from each of the numerator and de- 
nominator as you find have equal factors, and the products remain- 
ing to each term multiplied will he the true result. 



98 PRACTICAL ARITHMETIC. 

Thus, the first example above, 13 

7 *^ 7 *^^ 69 

0-X ^1 = YY The second, 7f bj 8f is -y and -^^ 

Then, \3 X f = 8^7. 2 

9 and 9 cancel eacn other. 14)897(64Jj Answer. 

84 
* 4 divides 52 and 8. 57 

1^ 



SUBTRACTION OF VULGAR FRACTIONS. 

Rule. — Reduce the fraction to a common denominator ; then sub- 
tract the less numerator from the greater^ and set the difference 
over the common denominator. 

Exception. — If tlie lower numerator be greater than the upper, subtract it from | 
the amount of the upper numerator and the common denominator added, and carry 
1 to the unit's place of the whole number. 

Note. — When the fractions are of different denominations, they have to be reduced, i 
to their proper value before they can be subtracted. 1 



. From i 


take |. 


« 


5)9 6 

3 2 

2 




,6 
3 

18 comn 


. From II 


'- take f|. 


4)88 

22 
9 


36 
9 


198. 
4 





9)18 = 2 X 7 = 14 
6)18 = 3 X 4 = 12 



Reduced ^)^ is I Ans. 



88)792 = 9 X 42 = 378 
' 36)792 = 22 X 21 = 462 

No. 2 presents an absurdity, as you 
at once perceive that 462 cannot be 

792"common denominator. subtracted from 378 without borrow- 

Au vv^xiaiiivix y^^i±yjLi^i.ii.<»xjyj ^^^ ^^ which the questiou does not 

have or allow. Then, suppose we say, from 21-36ths take 42-88ths, and it can be done, 
as 21-36ths is over one-half and 42-48ths is less ; so we may take less from more, but 
not more from less. 



f PRACTICAL 'ARITHMETIC. 



99 



3. From fi take 
4)36 88 
9 22 
22 

198 


If. 


36)792 = 22 X 21 = 462 
88)792 = 9 X 42 = 378 


4 

792 common 


denominator. 


or i|)^fe Ans. 



DIVISION OF VULGAR FRACTIONS. 

Rule. — Prepare the fractions^ if necessary : invert the divisor 
and multiply the numerators together for a new numerator, and the 
denominators for a new denomiiiator, and reduce to its lowest terms. 

Note. — If the fractions have a common denominator, divide the numerator of the 
dividend by the numerator of the divisor. 



1. Divide H by |. 
^. Divide iff by ii,. 



y IV 85 

^ 21 ~ F4' 



120 y 125 =- 16000 
T2 ^ 480 3I560' 



Note. — We find in these examples the quotient is greater than the dividend, which 
is always the case when the divisor is less than 1, but if the divisor be more than 1 
the quotient will be less than the dividend. 



3. Divide ^^ by |. 


f X 3^ = H reduced is f. 


4. Divide i by f . 


f X 1 = ^ = |Ans. 


5. Divide ^ by |. 

1 X ,\ = M. 


22)45(2^ Ans. 
44 


6. Divide -^^ by |. 


f X A = 1^ = t Ans 


7. Divide J of 1 by i of }. 

J of J = A. 

^ of J = ,V 
Then, ^ by ^ = 1^ X ^ = 


24)98(4J^ Ans. 
96 

98 2)^1 15 


8. Divide 1 by 4. 


i X 1 = ^ = 1 Ans. 


9. Divide 6f by J. 

5 

T3=1X 3,3 = IS. 


5)132(26f Ans. 
10 

30 



100 PRACTICAL ARITHMETIC. 

10. Divide | by J. 8)56(7 Ans. 



3_50 
50 • 



11. Divide STJ by 12 J. ^ XH= 

«7J 12i 50)350(7 Ans. 



'F 



The last two examples prove each other. Thus, f is 871^, and 871^ is |^; ^ is 12 J, > 
and 12J is ^ : hence the answer in each example is 7 times l-Sth. 



DECIMAL FRACTIONS. 

Note, — The decimal fraction is merely a different mode of expressing fractions 
denoted by a point to the left of a figure or figures, and the given decimal is the nume- 
rator ; the denominator is always lOths, lOOths, lOOOths, &c. according to the num- 
ber of figures in the decimal. .1 is J^, .15 is^^^, i^^'? ^^^ ^° °^» although ciphers 
placed after decimal figures neither increase nor decrease their value: thus, .1, .10, 
.100, .1000, they express in value the same, 1-lOth. But if ciphers are placed be- 
tween the decimal point and any other figure, the value is decreased in a tenfold pro- 
portion; as, .01 is Y^i -001 is j^, and so on ; and this decrease of decimals, counting 
from the left to the right, lessens value or quantity in a tenfold proportion, while count- 
ing from the right to the left increases in the tenfold proportion : hence, from thfse 
various reversions, the mathematician comprehends the philosophy of the science. 
And, learner, your advance must likewise come. 

Now, the operation in decimal fractions is simple and easy, and in expressed con- 
struction the same as the vulgar ; and when you express a decimal you should double 
use it ; as, .5 is 5-lOths, and 5-lOths is J; .2 is 2-lOths, 2-lOths is l-5th, and l-5th is 
.2 ; .20 is 20-lOOths, and 20-lOOthg is l-5th, and l-5th is .20 ; and so on. 



ADDITION OP DECIMALS. 

Rule. — Place both the whole numbers and the decimal numbers 
units under units, tenths under tenths, ■ hundredths under hun- 
dredths, and so on. 

1. Add 5.1, 80.71, 209.460, 8751.0214 together. 

5.1 

80.71 

209.460 

8751.0214 



9046.2914 
2. Add 64.28, 790.1, 3.01, 475.1, 2.231 together. 



PRACTICAL ARITHMETIC. 101 

64.28 

790.1 

3.01 
475.1 

2.231 



1334.721 



SUBTRACTION OP DECIMALS. 

Rule. — Place the less number under the greater^ so that units, 
tenths, and hundredths shall stand each und,er the other, right and 
left, each from the decimal point. 

1. From 90.2 take 19.48. 2. From 4000 take 999.001. 

90.2 4000 

19.48 999.001 



70.72 3000.999 

Note. — Right and left are whole numbers counted to the left, decimals to the right, 
each from the decimal point. 



MULTIPLICATION OP DECIMALS. 

Rule. — Place the numbers as in Simple Multiplication, and point 
off^ from the right hand as many figures as there are decimal Jig ures 
in both terms ; that is, in the multiplier and multiplicand. 

1. Multiply 24.15 by 18.14. 2. Multiply 4560 by .63. 

24.15 4560 

18.14 .63 



96 60 
2415 
19320 
2415 

438.0810 

3. Multiply .6012453 by .00008. 

.6012453 
.00008 



13680 
27360 

2872.80 



.000048099624 



102 



PRACTICAL ARITHMETIC. 



DIVISION OF DECIMALS. 

Rule. — Divide as in Simple Division, and point off as many 
figures from the right hand of the quotient as the decimals in the 
dividend exceed those of the divisor. 

Note. — If the number of figures in the divisor exceeds those of the dividend, annex 
as many ciphers as are necessary, and if the quotient does not have a sufficiency of 
decimals, add to the left hand of the quotient as many as are necessary. 



Divide 186513.239 by 304.81. 
304.81)186513.239(611.9 
182886 

36272 

30481 



Divide .00008757 by 35. 

35.).00008757(.00000250 
70 

175 
175 



57913 
30481 

274329 
274329 



7 rem. 



To reduce a vulgar fraction to a decimal. 

Rule. — Place ciphers to the right of the numerator, and divide hy 
the denominator until there is no remainder left, or until it is carried 
to a sufficient number of decimal places. 



1. Reduce J to a decimal. 

8)7000 



2. Reduce | to a decimal. 



.875 



3. 



4. 



5)40 



Reduce ^ to a decimal. 

16)90000(.5625 or b^. 
80 



100 
96 



4)100 

.25 

5. Reduce J to a decimal. 
8)1000(.125 or 12^ 
8_ 

20 
16^ 

40 
40 

To reduce any given sum or quantity to the decimal of any higher 
given denomination. 



40 
32 

80 
80 



PRACTICAL ARITHMETIC. 108 

Rule. — Reduce the given sum to the lowest denomiJiation men- 
tioned in it, and the denomination of which you ivish it a decimal 
to the same, and divide the given quantity hy it. 

Reduce 4s. %d. to the decimal of a pound. 

4s. M. £1 Then, 240)56000(.233 

12 20 480 



^cj 20 800 

oorf. ^2 720 



240(^. 800 

This would never end. 720 

Reduce 3 R. 8 per. to the decimal of an acre. 

R. per. 1 acDi. 

8 8 4 Then, 160)1280(.8 

4^ "4 roods. 12^^ 

40 

128 per. This .8 decimal is ^ of an 

160 perches. acre, or f. 
Reduce 1 pk. 2 qts. I pt. to the decimal of a bushel. 

pk. qts. pt. 1 bus. 64)21000(.328 

12 1 _4 192^ 

8 4 pecks. "180 

— 8 128 



10 qts, 

2 



82 quarts. 520 

2 512 



21 pts. ^^ P^s- ^ ^6^- 

To reduce a decimal fraction to its jjroper value. 

Rule. — Multiply the given decimal by the next loiver denomination, 
and cut off as many figures to the right hand as there are decimals 
in the given sum, and the remainder by the next lower denomination, 
and so on, until it is completed. 

1. What is the value of .575 of a dollar ? 
.575 

100 cents $1. 

r-rj rnn CtS. mills. 

10 mills 1 cent. 
5.000 



104 



PRACTICAL ARITHMETIC. 



2. What is the value of .826 of a yard ? 

' .826 

4 qrs. 1 yard. 
Vqq^ qrs. nails. 

4 na. 1 qr: 

T2I6 

3. What is the value of .360 of a day ? 

.360 
24 hours 1 day. 



Ans. 3 1.216 



1440 

720 



h. min. sec. 
Ans. -8 38 24 



8.640 

60 minutes 1 hour. « 

3"8l00 . 

60 seconds 1 minute. 

2T0OO 

Note. — For the Rule of Three in Decimals the question is stated as the Rule of 
Three in whole nombers, observing, when you multiply and divide, to place the decimal 
points according to the rules of multiplication and division of decimals. 



If 1.4 yds. cost 
Yds. Yds. 
1.4 : 15.9 : 
3.20 



1000 



3180 

47T 



J3.20, what will 15.9 yds. cost ? 

J 1.4)50,880(136.34.2.^ 

% 

■ ?L 

48 
42 ■ 

60 

50.880 ^Q 

"40 

28 

120 
112 

80 
70_ 

100 

98 



To reduce different denominations to a common denominator may be performed by 
multiplying each numerator into all the denominators except its own for a new nume- 
rator, and all the denominators into each other for a common denominator. 



PRACTICAL AIIITHMETIC. 



10; 



Thus, Reduce ^^g, J, and ^^ to a common denominator. 



3 


X 8 X 
12 

96 
3 

288 


12 : 


7 


X 12 X 


: 16 




i 






84 






16 






504 






84 





288 numerator for 



TS- 



1344 numerator for I. 



1344 



5 X 8 X 16 : 


= 640; 


numeraior 


for 


,%. 


8 










128 





















640 




■ 






for ^. 










Now, the denominators 






are, 16 


X 8 X 
12 

96 
16 

576 


12 







Then, we have ^-,, f|||, ,«^. 



1536 com. denominator 



To inform the learner of the complete harmony existing between 
decimal and vulgar fractions, we present another example as an illus- 
tration : Say 62i bushels of grain @ 43| cents per bushel. 

2d. Decimally. 

62.50 bushels. 



1st. By Multiplication. 
62J bushels. 
43f cents. 

186^ 

248 

31f 

15| 

21| 

$27.34f 

3d. Or both decimally. 

62.50 
43.75 
^ 31250 
43750 
18750 
25000 



43f cents. 



18750 
25000 
3125 



1562J 



= 3 

8- 



$27.3 

4th. Or, again, decimally and 
fractionally. 62.50 bushels. 
7 

■leths. 

16)43750(27.341 

32 



117 
112 



$27.34^^ 



r_5o_ 

000 



or f. 



55 

48_ 

70 
6J^ 

DM! 



106 



PEACTICAL ARITHMETIC. 



5th. Or, again, by Decimals 
and Practice. 



6th. Or, again, by Single Rule 
and Fractions. 



25 
12J 



IS 



IS 

61 is 



62.50 decimal bus. 



43| 



15.62.5 

7.81.21 
3.90.61 



621 



$2T.84aM or f . 



Now, they all produce the same re: 
suit; yet the last is the preferable 
method, though the learner should 
study them all. 



2 125 
4 175 

8 "625 

875 
125 

8)218.75 
$27.34f 



: 43i 
4 

175 



1. When cents are multiplied by cents, their product will be less 
than mills ; as, 50 cents by 50 cents is 2500 tenths of mills. 

Reduced is 



Reduced 10)2500(250 mills. 
20 



50 
50 







10)250(25 cents. 
20 

50 



2. Multiply 25 cents by 25 cents. 

25 cents. 

125 

6^5 or 6 cts. 2 mills and j^, or 6 J cts., or 6 ^ cts. 

3. Multiply 40 cents by 40 cents. 

40 Reduced, 10)1600 



1600 



160 mills or 16 cents. 



4. Multiply 6J cents by 6J cents. 

4 4 The 6^ decimal and 6.25 decimal. 

^ ^ X ¥ = «^«. 16)625(39^-10ths of mills. 

■ Or decimally. "*" 

6.25 145 

^ '6.25 144 

3125 1, 

125 10)39J5(3.9iVl0ths,or 3 m. 9Jg-10ths. 

3750 30 



39ff 25_i.educed is 3 m. 9^-lOths. 9J5-10ths. 



PRACTICAL ARITHMETIC. 



107 



5. Multiply 50 cents by 7 cents. 

7 

8j%o mills. 
10)350(35 mills. 
30 

"50 
50 



Reduced. 

10)35(3J cents. 
80 






Observation. — The above examples, by a little thought, are rendered plain, as 
in No. 5, 50 cents by 7 render their product tenths of mills, &c., as 50 cents X 25 
cents, or any other number of cents under 100 cents, are in sense and construction 
only so many parts of a hundi-ed. 50 cts. is 50-lOOths, or ^; 25 is 25-lOOths, or \; and 
so on. Now, if the 50 is 50-lOOths, the 7 is 7-lOOths : then 50-100ths X 7-lOOths = 
350-lOOths, and 350-lOOths is 3^ cents. So is the interest, discount, &c. of 50 cents 
at 7 per cent. 3.} cents, or 3 cts. 5 mills. This fully explains the cuUin<j ojf process 
in Simple Interest. 

This table shows the number of days from any day in one month to 
the same day in another month. 



From 
to 


January. 
February. 


^ 
^ 

^ 


< 


i- 
§ 






+3 

OQ 

<1 


1 

m 


c 

o 

o 


B 
O 


a 

o 
<v 


Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 


365 31! 59 90 1 120 


151 


181 212 1 243 273 304 


334 


33413651 28' 59! 89 


120 1 150 1 181 ! 212 ; 242 j 273 ! 303 


306 i 337 j 3651 311 61 


92 122 1 153 1 184 1 214 ! 245 276 


275 1 306 i 334 1 365 1 30 ! 61 1 91 122 1 153 183 | 214 | 244 


245 276 ! 304 ; 335 : 365 311 61 92 1 123 j 153 1 184 214 


214 1 245 1 273 I 304 1 334 1 365 1 30 61 1 92 1 122 153 


183 1 


184 215 ! 243 1 274 ! 304 335 365 


31 1 62 92 1 123! 153 


153 1 184 1 212 1 243 ! 273 1 304 1 334 1 365 1 31 ! 61 92 


122 1 


122 i 153 181 212 1 242 ' 273 ! 303 ! 334 365 30 i 61 1 91 ! 

1 III! 1 1 1 


92 1 123 1 151 i 182 1 212 ! 243 i 273 1 304 1 335 3651 31 1 61 


61 92 i 120 1 151 i 181 j 212 ! 242 ! 273 1 304 334 365 | 30 


31! 62 1 90 1 121 ! 151 ! 182 ! 212 ! 243 274 | 304 1 335 


365 



Take any month at the left hand, and any other at the top, and the 
number directly opposite the left-hand month and under the top month 



108 



PRACTICAL ARITHMETIC. 



is the time. Say the 15th September to the 15th August : the days 
are 334; or run September through from 15th Sept. to 15th January, 
122 days, February 153 days, March 181, April 212, May 242, June 
273, July 303, August 334, Sept. 365, Oct. 30, Nov. 61, Dec. 91 ; 
and so on, with* any of the months. 

Explanation. — Our plan of days work on p. 15. (Gr.) In regard to writing, children 
will soon learn to write by seeing the teacher at the blackboard and cutting letters in 
the air with a straw. In connection with his writing on the blackboard, call the letter 
to be cut, (the teacher cutting in the air as he walks around to see the performance 
of the pupils also.) So soon as the pupil is competent to copy the bills, notes, and 
other examples, he or she should copy them off. Those pupils who are in arithmetic 
of course will engage at it when not otherwise engaged ; and the advanced pupils, so 
soon as they have recited their regular morning lessons, will attend to book-keeping, 
&c. The teacher may vary his evening exercises as he may wish, differing in order] 
from the programme ; yet he should take each of them once a week. This work in-i 
eludes the exercises for four evenings, giving one evening to other exercises to the' 
choice of the teacher. 



POSITION. 

Teacher. What is Position ? 

Student. Position is a rule in which supposed numbers are used to 
obtain true results of such questions as cannot be solved by the com- 
mon rules of arithmetic. 

T. How many kinds ? 

S. Two, Single and Double. 

T. When is Single Position used ? 

S. When we use only one supposed number. 

T. How do you begin ? or what is first to be done ? 

S, Take a supposed number and work it according to the construc- 
tion, and then state as in Single Rule. 

T. How will you state it ? 

S. Take the result of the supposed number for the first, the sum 
supposed for the second, and the given sum for the third. 

T. Then what is to be done ? 

S. Work as the Rule of Three directs. 



A merchant being in debt, says |, |, |, and ^V of what he owes is 
690 dollars. How much did he owe ? Suppose he owed $972. 

972 I $ $ 

558.90 : 972 : : 690 
690 



194.40 

162 

121.50 

81 



8748000 Add 00 to 
5832 second term. 



558.90 result. 



67068000 



PRACTICAL ARITHMETIC. 109 

558.90)67068000($1200 Proof. 



55890 

111780 \ 

111780 \ 

00 ^'^ 



1200 



240 
200 
150 
100 

$690 



2. A person lent a sum of money at 7 per cent., simple interest, 
and at the end of 4 years and 6 months he received $350 interest. 
What amount did he loan ? 

Suppose he lent «800. $252 : 800 : : 350 
800 350 . 
T 40000 
2400 



5600 1 year. ' 252)280000(llll.ll.li 

252 



4i 



22400 4 years. 280 

2800 J or 6 months. ^^ 

280 

252.00 interest. 252 



280 

252 



280 

252 
280 
252 

^80 

252 

28 — 1 
55^ 9 

Note. — The learner is greatly benefited in the proof of a question, and the teacher 
should in every instance, when lecturing and solving a question, avail himself of the 
opportunity of the proof of the theory in each instance, as well as in the arithmetical 
department ; and in each instance in which a question is proven, a double use and 
instruction is attained. Therefore we give the proof of the last example, as more 
science is required in the proof than there is in the solution. 



110 PRACTICAL ARITHMETIC. 

The amount loaned, $llll.ll.li 

7 - 



6 mos. J 



1711. 11. 1^ for 1 year. 



4 I X 4 = f = 3^, add the 3. 



31111.11.11 for 4 /ears. 

3888.88.8| and 1 over and | is 1| = V- 



$35000.00.0 



1| is V^, and the half of \^ is |. 
9 



DOUBLE POSITION. 

Teacher. How do you first proceed in Double Position ? 

Student. Suppose a number and work it according to the construc- 
tion in the question, and subtract the result or the given amount one 
from the other, which will give the first error. 

T. How will you proceed after finding the first error ? 

S. Suppose some other number and work in the same way, which 
will give the second error. 

T. After having obtained both errors, what is to be done ? 

S. Multiply the first error by the second supposed number, ancL 
the second error by the first supposed number. 

T. What is then to be done ? 

S. If the errors are both too great or both too little, divide the dif- 
ference of the products by the difierence of the errors. 

T. Should there be one too great and the other too little, how will you proceed ? 

aS'. If the errors are one too great and the other too little, divide 
the sum of the products by the sum of the errors. 

A farmer sold his cattle, hogs, and sheep for $1215, being paid at 
the rate of $9 per head of cattle, $6 per hog, and $3 per sheep. 
There was a certain number 5f hogs, as many cattle as hogs save 10, 
and half as many sheep as cattle. How many did he sell of each ? 

First, suppose 60 hogs @ $6 each, $360 
Cows off, 10 

50 cows @ $9 each, 450 
Half as many sheep, 25 sheep @ $3 each, 75 

Whole amount sold, 885 
1215 



330 error too little. 



PRACTICAL ARITHMETIC. 



Ill 



Then, first error by Again, 70 hogs @ $6, $420 
2d supposed number. 10 off. 

330 — 

70 60 cows @ $9, 540 

Half as many, 30 sheep @ $3, 90 



23100 
9900 

13200 difference. 

First error, 330 

Second '' 165 



Whole amount, 1050 
1215 



Difference, 165 
Difference of products. 



165 error too little. 

Then, second error by first 
supposed number. 
165 
60 



165)13200(80 hogs. 
1320 







80 hogs @ $6. 
6 



80 hogs. 
10 off. 

70 cows. 
35 sheep. 

Proof. 



9900 



35 sheep 



$480 hogs. 

70 cows @ $9. 
9 



$105 sheep. 
630 cows. 
480 hogs. 



$630 cows. 



$1215 all. 



A company of 4 persons enter copartnership in trade to the effect 
that their gain shall be divided so as to give A a certain part, B 
$24 more than A, C $14 less than B, and D as much as A and C 
together. Their gain was $1044. What is each partner's share ? 



112 



PRACTICAL ARITHMETIC. 



Suppose A to have §100 Suppose A 



24 more. 



B, $124 

14 less. 



25000 

500 

75000 



24 



B, 



.74 
14 



75000 

25000 

250)50000($200 A's. 
500 24 



c, no 


C, 160 


00 


224 B's. 


100 


150 




14 


D's, 210 


D's, 310 




210 C's. 


C's, no 


C's, 160 




•200 


B's, 124 


B's, 174 






A's, 100 


A's, 150 




410 D's. 

210 C's. 


All, 544 


794 




224 B's. 


1044 


1044 




200 A's. 


500 too little. 


250 too little. 




1044 


150 


100 


Sum of gain. 



25000 



ANNUITIES AT SIMPLE INTEREST. 



Teacher. What are Annuities ? 

Student. Pensions, salaries, rents, or any sum of money payable 
yearly for a number of years or forever. 

T. If the annuity is not paid yearly, how is interest computed ? 

S. If the annuity is not paid yearly, each year's interest draws 
interest from the time the interest is due. 

T. What is the rule for annuities unpaid for several years ? 

S. Find the interest of the given sum for 1 year at the given rate 
per cent. ; multiply this interest by the number of years, less 1 ; •add 
double the given sum to this product, and then multiply this sum by 
half the number of years. 

NoTE.^ — The reason that the interest for 1 year is multiplied by 1 year less than the 
number of years to obtain the interest, is because interest is not due the first year. 

What is the amount of a pension, salary, house-rent, or any income 
payable yearly, of $500, which has been due 20 years, at 5, 6, and 7 
per cent. ? 



PRACTICAL ARITHMETIC. 



113 



§500 at 5 per cent. • ?500 at 6 per cent. $500 at 7 per cent. 
5 6 7 



25.00 1 year. 
19 years. 



22500 
2500 



80.00 1 year. 
19 years. 



27000 
3000 



35.00 1 year. 
19 years. 



31500 
3500 



§475.00 for 19 years. $570.00 for 19 years. $665.00 for 19 years. 
1000 double amt. 1000 double amt. 1000 double amt. 



1475 $1570 $1665 

10 half no. of yrs. 10 half no. of yrs. 10 half no. of yrs. 



14750 dollars. 



15700 dollars. 



16650 dollars. 



To find the present worth of an annuity for an unlimited time. 

Teacher. Will you repeat the rule for an unlimited annuity ? 

Student. Annex five ciphers to the given annuity, and divide by 
the given rate per cent., and the quotient will be in mills; then cut off 
one figure for mills and two for cents : the remainder will be dollars. 



What is the present worth of a salary, income, rent, pension, 
estate worth $900 a year, at 6, 7, and 8 per cent. ? 

At 8 per cent. 
8)90000000 

$11250.00.0 



or 



At 6 per cent. 
6)90000000 

$15000.00.0 
6 

Proof $900.00.000 



At 7 per cent. 
7)90000000 

$12857.14.2t 
7' 

$900.00.000 



$900.00.000 



To find tJie present ivorth of an annuity which is to continue a 
limited or given time. 

Teacher. Will you repeat the rule for a limited annuity ? 

Student. Annex five ciphers to the given annuity, and divide by the 
amount of $1 for one year ; for two years, divide it by the amount 
of $1 for two years, and so on, for three, four, or more years, and 
add their quotients together. 



114 PRACTICAL ARITHMETIC. 

What is the present worth of a salary, pension, rent, income, &c. 
of $800, to continue 3 years at 6 and 7 per cent. ? 

$1 at 6 per cent, for 2 year* $1 at 6 per cent, for 1 year is 

is $1.12. $1.06. 

112)80000000(714.28.5if 2 yrs. 106)80000000(754.71.6|| 1 year. 

784 742 714.28.5i| 

"160 580 677.96.6^ 

112 . 530_ $2146.96.8f|| 

"480 500 

448 424 



320 760 

224 742 



960 180 

896 106 



\104fS2 
/106\53 



640 740 

560 636 

sVso/'io 

1? + 11 = if- $1 at ^ per cent, for 3 years is 

H^l 18 
ft ^ ^ 118)80000000(677.96.6^ 3 yrs. 

708 



212 
53 

742 com. denominator. 



920 

826 



940 

53)742(14 X 52 = Iff. 826 

53 

21 
212 

)74 
70 



1140 
212 1062 



780 
14)742(53 X 13 = f||. 708 

"720 
42 708 



42 

689 _J_ SM = WJ = 1 §T§ 



f)*(A 



Note. — The learner need not think these added ciphers are remarkable or of dif- 
ficult solution: they are merely on the Single Rule or Proportion system, — two ciphers 
being necessary for dollars, two to the cents, and one for mills. The $900 or $800 in 
the last two examples is worked by the Single Rule in sense : as 6, 7, or 8 is to 900, 
so is 100 to both, which would give the dollars ; and annex ciphers to obtain the cents 
and mills. So with 106, 112, 118, &c. 



PRACTICAL ARITHMETIC. 115 



INVOLUTION. 

Involution is a process of multiplying a number by itself a cer- 
tain number of times. 

Rule. — 3Iult{ply the given number ly itself, and the product by the 
same nu77iber, and so on, until the number of times is exhausted. 

A general formed his army into a square, and found there were 65 
men in rank and Qb in file. How man}^ men had he ? 

65 
65 



325 
390 



4225 men. 

A field of corn has 180 rows and the same number of hills in each 
row. How many hills in the field ? 

180 
180 



14400 
180 



32400 hills. 



SQUARE ROOT. 

Teacher. How do you prepare the given sum iu Square Root ? 
Student. Separate the given number by dots into periods of two 
figures each, beginning at the right hand. 

T. After separating the given number, what is to be done ? 

>S'. See what number, multiplied by itself, will come the nearest to 
the left-hand period, without exceeding it, as a divisor. 

T. After having found this divisor, how do you proceed ? 

*S^. Place this number in the quotient and its square under the left- 
hand period, and subtract therefrom, then to the right hand of the 
remainder bring down the next period, (two figures.) 

T. Then how do you proceed ? 

S. Double the quotient figure for the next divisor, and try how often 
this divisor is contained in the dividend, omitting the right-hand figure. 



116 PEACTICAL ARITHMETIC. 

T. How then? 

/Sf. Place the figure signifying the number of times both in the 
quotient and to the right of the divisor. 
T. How then? 

&. Proceed in the same way until all the periods have been brought 
down. 

Note. — Decimals and fractions are worked in the same manner: in decimals, if 
there is only one figure at the last place, add a cipher to make an even period ; in 
fractions, square the numerator and denominator. 



1. What is the square root of 1225 ? 



3)1225(35 Ans. 
9 

65")325 
325 



2. What is the square root of 99980001 ? 
9)99986001(9999 Ans. 

Q1 O 



81 ^ 

189)1898 18 

1701 

1989)19700 
17901 

19989)179901 
179901 

3. An agent of a railroad paid $9801 to a number of men, and each 
man received as many dollars as there were men. How many men 
were there ? 

9)9801(99 men. •^^^'''^^ 

189)170118 -I? •^°"'^'^^- 

1701 ggl 



$9801 

4. A polygonal castle had on each side an equal number of cannon. 
The whole number was 1089, the number of cannon on each side 
being equal to the number of sides. How many sides ? and how many 
cannon on each side ? 



PRACTICAL ARITHMETIC. 117 

3)i089(33 sides, and 33 cannon to each side. 

9 Proof. 

63")T89 f 

189 ^ 

99 
99 

• 1089 

j^OTE. — The teacher can readily prepare questions for Ms pupils ; and for training 
he should use numbers that will come out even by squaring any number he wishes, as 
is shown in the proof of these examples, and then present the question accordingly. 

What is the square root of f|f^ ? 
Now, 2450)3200(1 Then, f^)fMt(|| = I- 



2450 

'2250 



750^2450^3 ■^^^' *^® ^^ ^^ reduced to its 

oof;n lowest terms, which is 



200)750(3 
600' 



49 = I. An«s 
61 8 ^nb. 



150)200(1 
150 



50)150(3 
150 

What is the square root of f f ? If = I -^^s. 

What is the square root of .002916 ? 

6029i6(.054 

25 
104)416 
416 

Note. — If the dividend is too small, place ciphers in the quotient, also one in the 
divisor. 

CUBE ROOT. 

Teacher. How do you prepare a question in the Cube Root ? 

Student. Separate the given number by dots into periods of three 
figures each from the right hand. 

T. How will you then proceed ? 

S. Find the greatest cube in the left-hand period as a divisor, and 
place its root in the quotient and its cube under the first period and 
subtract it therefrom, and to the remainder bring down the next period 
for a dividend. 

T. What then is to be done ? 



118 



PRACTICAL ARITHMETIC. 



S. Square the root or quotient, and multiply the square by 3 for a 
defective divisor. 

T. What is to be done with this defective divisor ? 

S. Try how often it is contained in the dividend, omitting the two 
right-hand figures, and place this number to the right of the quotient 
and its square to the right of the defective divisor : if the square be 
less than ten, supply the ten's place with a cipher. 

T. How will you* proceed to obtain a complete divisor ? 

S. Multiply the quotient by the last root and that product by 30 ; 
add the product to the defective divisor to complete it. 

T. How then ? 

S. Multiply and subtract as in Long Division, and bring down the 
next period, and so on, continually, until all the periods have been 
brought down. 

Note. — For extracting the cube root of vulgar fractions, it is best to bring them to 
decimals and proceed as in whole numbers. The result, of course, will be decimal. 



What is the cube root of 456533 ? 

7x7 = 49x7 = 343. 

J_ • 
49 
3 

147 defective divisor. 

7 
_7 
49 

30 

1470 product of 30 and square 

Defective divisor and square of 
7 = 14749 
1470 



456533(77 
343 



16219)113533 
113533 



Proof. 

77 
77 
539 
539 



by last quotient. 



16219 complete divisor. 



5929 

77 

41503 
41503 

456533 



By the proof you can readily make your own questions either in the square or cube 
root. 



ARITHMETICAL PROGRESSION. 



Teacher. What is Arithmetical Progression ? 

Student. Arithmetical Progression is a series of numbers which in- 
crease or decrease by a continual addition or subtraction of the same 
numbers. 



PRACTICAL ARITHMETIC. 119 

T. How many considerations are to be particularly attended to in Aiitiitactical 
Progression ? 

>§. Five, — viz. : the first term, the last term, the number of terms, 
the common difference, and the sum of all the terms. 

T. When the first term, common difference, and nmnber of terms are given to find 
the last term and amount of all the terms, what is the rule ? 

S. Multiply the number of terms less 1 by the common difference, 
and to their product add the first term for the last term, and add the 
first and last terms together and multiply by half the number of terms 
for the amount. 

1. A merchant sold 60 yards of lace at 2 cents for the first yard, 4 
for the second, and so on, increasing 2 cents every yard. What did 
the lace amount to ? 

60 yards less 1 = 59 
Common difference, 2 

118 

2 first term. 



120 last term. 



first term, 2 



122 sum. 

30 half number of terms. 



$36.60 Ans. 

2. A man travels 10 miles the first day, increasing 4 miles each 
day for 16 days. How many miles did he travel the sixteenth day ? 
and how many miles in all ? 

16 days less 1 = 15 

4 common difference. 

60 

10 first term. * 

Ans. the last day, 70 last term. 
10 

80 
8 half number of terms. 



640 miles in all. 

3. A farmer bought 100 sheep : for the first he paid 4 cents, the 
next 8, and so on, increasing 4 cents. What did the last sheep cost ? 
and what did all the sheep cost ? 



120 PRACTICAL ARITHMETIC. 

ICO less 1 = 99 

4 com. difference. 

396 Now, the learner must make the com- 

4 first term parison between Arithmetical and Geo- 

* metrical Progression, as the former increases 

Last sheep, 4.00 or decreases by a common addition or sub- 

^ traction, while the latter increases or de- 

creases with such rapidity as almost to seem 

404 improbable. 

50 half no. terms. 



$202.00 

GEOMETRICAL PROGRESSION. 

Teacher. What is Geometrical Progression ? 

Student. Geometrical Progression is the increase of a series of 
numbers by a common multiplier, or decrease by a common divisor. 

T. What is the rule ? 

;S^. Raise the ratio to the power whose index is one less than the 
number of terms given ; then multiply the product by the first term, 
and that product will be the last term ; then multiply the last term 
by the ratio, from the product subtract the first term, and divide the 
remainder by the ratio less 1 for the sum of the series. 

If a person at the birth of an heir should deposit 1 cent towards 
its portion, promising to double it at the return of every birthday 
until it should be 21 years of age, what is its portion ? 



Ratio, 



II 1 16 h h 128 256 512 
512 


1024 
512 

2560 


262144 18th power. 
4 2d power. 

1048576 20th power. 
1 first term. 


1048576 last term. 
2 ratio. 


2097152 

1 first term. 


1)2097151 



$20971.51 portion. 



t P K A C T I C A L A Tv I T II M E T I C. 121 

The learner may exercise on two examples, — the 60 yards of lace, in Arithmetical 
Progression, and the 100 sheep, by Geometrical Progression, from which he can form a 
correct estimate of the contrast in the philosophy of the two rules, &c. ; comparison is 
a good instructor. 



CONTENTS OP LAND MEASURED BY THE ROD. 

Rule. — Multiply the length by the breadth : if one end of the 
land is broader than the other, multiply the length by the medium of 
breadth, ^c. 

How many acres in a piece of land 168 rods long and 106 rod^ 
wide ? 



168 length. 
106 width. 



1008 
Divide by 160, because igg 

160 rods make an acre. 



160)17808(111^ Acres. 
160 

180 
160 



208 
160 

lJo\A3jA. . 
lb'l60VlO 

How many acres in a field 84 rods long and 53 wide ? 

84 long. 
53 wide. 

252 
420 



160)4452(27 A. 132 Rods. 
320 
1252 
1120 



132 rods. 

How many acres in a piece of land 142 rods long, and 72 at one 
end and 44 at the other end ? 



122 



PRACTICAL ARITHMETIC. 



72 
44 


Then, the land is 142 rods long 
and 58 broad. 


)28 difference. 
14 medium. 


142 long. 
58 broad. 



Now, to the medium add the 
narrowest end, or subtract the me- 
dium from the broadest, which gives 
the medium or breadth. 

44 
Add 14 

58 



or 


7^ 


off 


14 




58 



1136 
710 



160)8236(51,;^ or 76 rods. 
800 



236 
160 



4\ V6 (19 

I/TeoVlo 



A RIGHT-ANGLED TRIANGLE. 

Rule. — Multiple/ one hy half of the other ^ and reduce to acres, 

A triangular piece of ground is 40 rods base and the perpendicular 
is 30 rods. How much land ? 




Perpendicular 30, half is 15. 
15 
40 

160)600(3 A. 120 R. or }. 

480 
40Y120/3 

40/160\4 



Base 40, or half base 20. 
20 
30 

160)600(3} A. 
480 



toneoV 



Note. — If the land is measured by the yard, divide by 4840 to reduce to acres ; a8 
4 R. X 40 r. or per. X 30|^ yds. = 4840. Farmers can measure fields or diflferent 
parcels of land by the above rules. 



A CIRCLE OR WHEEL. 

To find the area of a circle. 
Rule. — Multiply half the circumference hy half the diameter. 

A gentleman has a circular garden or fish-pond, the diameter being 



PRACTICAL ARITHMETIC. 123 



112 feet and the circumference 352 feet. How many feet in the 
garden or fish-pond ? 

Diameter 112, half 56. 17( 

Circumference 352, half 176. 51 

1056 

880 

9856 square feet. 






To find the circumference of a circle or wheel. 
Rule. — Blultiply the diameter hy 22, and divide hy 7. 

What is the circumference of a wheel 36 feet in diameter ? 

36 

22 

72 
72 



7)7^ 
Circumference, 113^ feet. 

What is the diameter of a wheel, the circumference being 113{ feet ? 

113^ feet. 

7 
22)792(36 feet (and proof). 

132 
132 



BOARD OR SCANTLING MEASURE. 

Rule. — To find the contents of hoards^ multiply the length in feet 
hy the width in inches, and that product hy the numher of hoards ; 
then divide hy 12, and the quotient will he in feet and the remainder 
inches. 

Note. — This rule supposes the boards as one inch thick. If the boards are more 
than one inch thick, the edge is to be reckoned as in scantling or joist measure. 

How many feet in 60 plank 14 feet long and 15 inches wide ? 



124 



PRACTICAL ARITHMETIC. 



14 long. 

15 wide. 



14 



or 60 plank. 
14 long. 

240 
60 



A 



210 


840 


60 number plank. 


1 J 15 in. is lift. 


12)12600 


840 


1050 feet. 


210 




1050 feet. 


70 plank 12 feet long, 8 in. wide ? 


or 70 plank. 


12 

o 


12 long. 


8 


840 


96 
70 


f 8 in. is 1 ft. 

280 


12)6720 


280 



560 feet. 



560 feet. 



SCANTLING OR JOISTS. 

UvLE.— Multiple/ the length, side, and edge together, and their 
product hy the number of pieces ; then divide hy 12. 

How many feet in 80 scantling 12 feet lon^, 2 inches thick, 3 wide ? 
(That is, 2 by 3.) 

12 

or, 2 by 3 is 6, 6 

72 



12 length. 



24 

3 
72 

80 



80 



12)5760 feet. 
• "480 feet. 
The same 3 by 5. 
12 

36 
5 

i80 

80 



or, 3 by 5 is 15, 



12) 5760 

480 feet. 



15 
12 

l80 
80 



12)14400 

T200leet. 



12 )14400 
~l200~feet. 



PRACTICAL ARITHMETIC. 125 

To measure a crib. 

Rule. — 31ultiply the lengthy ividfh, and depth in feet together, 
and multiply their product by 8 and divide by the numh^of quarts 
to the peck. 

Note. — This is the most convenient rule for fai'mers, although it may lack a small 
fraction, but for the unevenness of log cribs it is as correct as any rule, and much more 
easily calculated. 

Say a crib 20 feet long, 12 wide, and 9 deep. How many bushels ? 

20 long. Or, tlie same at 9 qts. to the pk. 

_1? ^i^e. 2160 bus. at 8 qts. to pk. 

240 8 

9 deep. 9)17280 

2160 2)1920 bus. at 9 qts. 

^qts. 1 pk. arithmetically. ^^ ^^^^^^ 



10)17280 ^^2 barrels at 9 qts. 

2)1728 bus., that is, m cob. 
5)864 bus. it will shell. 
172| barrels. 

Note. — Measures vary from 8 to 10 qts. to the pk., 2 bus. in the cob "will shell 1 bus. 
5 bus. 1 bbl. 

A crib 12 feet long, 8 J wide, and 6 J deep? 

12 long. 
8J wide. 

96 
_^6 

102 

_3 

612 
51 

663 

8 

10)5304 

2)530 and ^ = I = 1 pk. 4 qts. If pts. 
5) 2651 bus. shell, (lo qts. to pk.) 
53^ barrels. 

To find the number of shingles necessary to cover a roof of any 
given dimensions. 
9 



126 PRACTICAL ARITHMETIC. 

Rule. — Multiply the length of plate hy the length of the rafter^ 
which will give the square feet on one side of the roof ; double it for 
the square feet on both sides; then multiply by the number of shingles 
to a square foot, and the product will be the number. 

How many shingles will be required to cover a roof, the plate being 
40 feet long and the rafters 15 feet long, the shingles 4 inches wide 
and show 6 inches ? 

Plate, 40 feet long. 
15 feet rafter. 



200 
40 



Now, shingles 4 incTies wide take 3 to 
make a foot in width. 3 X 4 = 12 in. ; 



600 feet on one side. ^^^ ^^ show 6 in. is only J foot: therefore 



2 



it will take twice 3 to show a foot, which 
is 6 shingles. 



1200 feet on both sides. 

6 shingles to the sq. foot. 
7200 shingles. 

Suppose the plate of a roof to be 120 feet long and the rafters 
30 feet, and the shingles 4 inches wide and are laid to show but 4 
inches. How many shingles will be required ? 

Length of plate, 120 feet. 
Length of rafter, 30 feet. 



4ft 



3600 feet on one side. 

2 



7200 on both sides. 

9 shingles to sq. foot. 



64800 shingles. 

Now, 4 inches wide will require 3 times 4 to make 1 foot wide ; and if 4 inches are 
only shown, it will require 3 times 4 to give a foot in length, 3 times 3 being 9, the 
number of shingles to a square foot. 



d^ 



-\)< 



wmmmM 




